UNIVERSITY  OF  CALIFORNIA. 


FROM  THE    LIBRARY  OF 

DR.  JOSEPH   LECONTE. 

GIFT  OF  MRS.  LECONTE. 
No. 


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s 


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L  I 


T. 


Light. 


4. 

Bodies   re- 
garded as 
self-lumi- 
nous and 
opaque. 


Opaque  bo- 
dies become 
luminous  in 
the  presence 
of  a  lumi- 
nous body. 


PART  I. 

Of  unpolarized  Light. 
§  I.   INTRODUCTION. 

IN    this  article  we  propose  to  give  an  account  of  the  properties  of  light ;  of  the  physico-mathematical  laws     Part  I. 
which  regulate  the  direction,  intensity,  state  of  polarization,  colours,  and  interferences  of  its  rays ;    to  state  the  v^ ~ v_^. 
theories  whicii  have  been  advanced  for  explaining  the  complicated  and   splendid  phenomena  of  optics;    to 
explain  the  laws  of  vision,  and  their  application,  by  the  combined  ingenuity  of  the  philosopher  and  the  artist,  to 
the  improvement  of  our  sight ;   and  the  examination  and  measurement  of  those  objects  and  appearances  which, 
from  their  remoteness,  minuteness,  or  refinement,  would  otherwise  elude  our  senses. 

The  sight  is  the  most  perfect  of  our  senses  ;  the  most  various  and  accurate  in  the  information  it  affords  us  ; 
and  the  most  delightful  in  its  exercise.  Apart  from  all  considerations  of  utility,  the  mere  perception  of  light  is 
in  itself  a  source  of  enjoyment.  Instances  are  not  wanting  of  individuals  debarred  from  infancy  by  a  natural 
defect  from  the  use  of  their  eyes,  whose  highest  enjoyment  still  consisted  in  that  feeble  glimmering  a  strong 
sunshine  could  excite  in  their  obstructed  organs  ;  but  when  to  this  we  join  the  exact  perception  of  form  and 
motion,  the  wonderous  richness  and  variety  of  colour,  and  the  ubiquity  conferred  upon  us  by  just  impressions  of 
situation  and  distance,  we  are  lost  in  amazement  and  gratitude. 

What  are  the  means  and  mechanism  by  which  we  receive  this  inestimable  benefit  ?  Curiosity  may  well  prompt 
the  inquiry,  but  a  more  direct  interest  urges  us  to  pursue  it.  Knowledge  is  power ;  and  a  careful  examination  of 
the  means  by  which  we  see,  not  only  may,  but  actually  has  led  us  to  the  discovery  of  artificial  aids  by  which 
this  particular  sense  may  be  strengthened  and  improved  to  a  most  extraordinary  degree ;  giving  to  man  at  once 
the  glance  of  the  eagle,  and  the  scrutiny  of  the  insect — by  which  the  infirmities  of  age  may  be  deferred  or 
remedied — nay,  by  which  the  sight  itself  when  lost  may  be  restored,  and  its  blessings  conferred  after  long  years 
of  privation  and  darkness,  or  on  those  who  from  infancy  have  never  seen.  But  as  we  proceed  in  the  inquiry 
we  shall  find  inducements  enough  to  pursue  it  from  purely  intellectual  motives.  A  train  of  minute  adaptation 
and  wonderful  contrivance  is  disclosed  to  us,  in  which  are  blended  the  utmost  extremes  of  grandeur  and 
delicacy;  the  one  overpowering,  the  other  eluding,  our  conceptions.  In  consequence  of  those  peculiar 
and  singular  properties  which  are  found  to  belong  to  light  in  its  various  states  of  polarization,  it  affords 
to  the  philosopher  information  respecting  the  intimate  constitution  of  bodies,  and  the  nature  of 
the  material  world,  totally  distinct  from  the  more  general  impressions  of  form,  colour,  distance,  &c. 
which  it  conveys  to  the  vulgar.  Its  notices,  it  is  true,  in  this  respect,  are  addressed  rather  to  the  intellect 
than  the  sense  ;  but  they  are  not  on  that  account  less  real,  or  less  to  be  depended  on.  Polarized  light 
is,  in  the  hands  of  the  natural  philosopher,  not  merely  a  medium  of  vision  ;  it  is  an  instrument  by  which  he 
may  be  almost  said  to  feel  the  ultimate  molecules  of  natural  bodies,  to  detect  the  existences  and  investigate  the 
nature  of  powers  and  properties  ascertainable  only  by  this  test,  and  connected  with  the  more  important  and 
intricate  inquiries  in  the  study  of  nature. 

The  ancients  imagined  vision  to  be  performed  by  a  kind  of  emanation  proceeding  from  the  eye  to  the  object 
seen.  Were  this  the  case,  no  good  reason  could  be  shown  why  objects  should  not  be  seen  equally  well  in  the 
dark.  Something  more,  however,  is  necessary  for  seeing  than  the  mere  presence  of  the  object.  It  must  be  in  a 
certain  state,  which  we  express  by  saying  that  it  is  luminous.  Among  natural  bodies  some  possess  in  themselves 
the  property  of  exciting  in  our  eyes  the  sensation  of  brightness,  or  light ;  as  the  sun,  the  stars,  a  lamp,  red-hot 
iron,  &c.  Such  bodies  are  called  self-luminous ;  but  by  far  the  greater  part  possess  no  such  property.  Such 
bodies  in  the  dark  remain  invisible,  though  our  eyes  are  turned  directly  towards  them  ;  and  are  therefore  termed 
dark,  non-luminous,  or  opaque,  though  this  word  is  also  used  occasionally  to  express  want  of  transparency.  All 
bodies,  however,  though  not  luminous  of  themselves,  nor  capable  of  exciting  any  sensation  in  our  eyes,  become 
so  on  being  placed  in  the  presence  of  a  self-luminous  body.  When  a  lamp  is  brought  into  a  dark  room,  we  see, 
not  only  the  lamp,  but  all  the  other  bodies  in  the  room.  They  are  all,  so  long  as  the  lamp  remains,  rendered 
luminous,  and  are  in  their  turn  capable  of  illuminating  others.  Thus  a  sunbeam  passing  into  a  darkened  room 
renders  luminous,  and  therefore  visible,  a  sheet  of  paper  on  which  it  falls  ;  and  this,  in  its  turn,  will  in  like 
manner  illuminate  the  whole  apartment,  and  render  visible  every  object  it  contains,  so  long  as  it  continues  to 
receive  the  sunbeam.  The  moon  and  planets  are  opaque  bodies ;  but  those  parts  of  them  on  which  the  sun 
shines  become  for  the  time  luminous,  and  perform  all  the  offices  of  self-luminous  bodies.  Thus  we  see,  that  the 
communication  which  we  call  light,  subsists  not  only  between  luminous  bodies  and  our  eyes,  but  between 
luminous  and  non-luminous  bodies,  or  between  luminous  bodies  and  each  other. 

2  Y* 


342  LIGHT. 

Light.          Many  bodies  possess  the  property  of  intercepting  this  peculiar  intercourse  between  luminous  bodies  and  our     Part  L 
v— >-v-»»'  eyes,  or  other  bodies.     A  screen  of  metal  interposed  between  the  sun  and  our  eyes  prevents   our  seeing  it ; 

5.  interposed  between  the  sun  and  a  sheet  of  white  paper,  or  other  object,  it  roste  a  shadow   on  such  object :    t.  e. 
Opaque  bo-  renders  it  non-luminous.     By  this  power  of  bodies  to  intercept  light,  we  learn  that  the  communication  which 

BS  '."'?'/'    constitutes  it  lakes  place  in  straight  lines.     We  cannot  see  through  a  bent  metallic  tube,  nor  perceive  the  least 
glimpse  of  light  through   three  small   holes  in  as  many  plates  of  metal   placed  one  behind  the   other  at  a 
distance,  unless  the   holes   be  situated  exactly  in  one  straight  line.     Moreover,  the  shadows  of  bodies,  when 
Light  ema-  fairly  received  on  smooth  surfaces  perpendicular  to  the  line  in  which   the  luminous  body  lies,  are  similar  in 
te.s  ln        figure  to  the  section  of  the  body  which  produces  them,  which  could  not  be,  except  the   light  were   commu- 
nicated in  straight  lines  from  their  edges  to  the  borders  of  the  shadow.     We  express  this  property  by  saying 
that  light  emanates,  or  radiates,  or  is  propagated  from  luminous  bodies  in  straight  lines  ;  by  which  expressions 
nothing  more  is  to  be  understood  than  the  mere  fact,  without  in  any  way  prejudging  the  question  as  to   the 
in  all  di-      intimate  nature  of  this  emanation.     Moreover,  it  emanates  from  them  in  all  directions,  for  we  see  them  in  all 
rections,       situations    of  the  eye,  provided    nothing    intervene    to  intercept  the  light.      This  is  the  essential   distinction 
between   luminous  bodies  and   optical   images ;    from  which,  as  we  shall  see,  light  emanates   only  in  certain 
directions.     Whether  it  emanates  equally  in  all  directions  will  be  considered  farther  on. 

6.  Light  also  radiates  from   every  point  (at  least  from  every  physical  point)  of  a  luminous  body.     This   may, 
and  from      perhaps,  be  regarded  as  a  truism  ;    for  those  points  of  a  luminous  body  from  which  (as  from  the  spots  in  the 
Every  physi-  sun)  no  ]jg.nt  emanates,  are,  in  fact,  non-luminous,  and  the  body  is  only  partially  so ;  the  figure  of  the  spots  is 
aluminous"  on'y  seen>  Decause  it  is  a'so   necessarily  that  of  the  luminous  surface  which  surrounds  them.     Still  it  should 
surface.        be  borne  in  mind,  for  reasons  which  will  appear  when  we  come  to  speak  of  the  formation  of  images.    It  is 

possible  (nay,  probable)  that  a  luminous  surface,  such  as  that  of  the  flame  of  a  candle,  may  consist  only  of  an 
immense  but  finite  number  of  luminous  points,  surrounded  by  non-luminous  spaces ;  but  it  is  not  ocular 
demonstration  this  idea  admits  of;  and  it  is  sufficient  for  our  purpose  that,  so  far  as  our  senses  inform  us,  every 
physical  point  of  a  luminous  surface  is  a  separate  and  independent  source  of  light.  We  may  magnify  in  a 
telescope  the  sun's  disc  to  any  extent,  and  intercept  all  but  the  very  smallest  portions  of  it,  (spots  excepted,) 
yet  the  visibility  of  one  part  is  no  way  impaired  by  the  exclusion  of  the  rest.  In  this  sense  the  proposition 
is  no  truism,  but  an  important  fact,  of  which  we  shall  hereafter  trace  the  consequences. 

7.  When  the  sun    shines    through    a    small    hole,    and    is    received    on    a   white   screen    behind    at    a    con- 
siderable distance,  we  see  a  round  luminous  spot,  which  enlarges  as  the  screen  recedes  from  the  hole.      If 
we  measure  the  diameter  of   this  image  at  different  distances  from  the  hole,    it  will    be  found    that  (laying 
out  of  the  question  certain  small  causes  of   difference  not  now  in  contemplation)  the    angle    subtended    by 
the  spot  at  the  centre  of  the  hole  is  constant,  and  equal  to  the  apparent  angular  diameter  of  the  sun.     The 
reason  of  this  is  obvious ;  the  light  from  every  point  in  the  sun's  disc  passes  through  the  hole,  and  continues 
its  course  in  a  right  line  beyond    it  till  it  reaches  the  screen.      Thus    every  point    in    the    sun's  disc  has  a 
point  corresponding  to  it  in  the  screen ;  and  the  whole  circular  spot  on  the  screen  is,  in  fact,  an  image  or 
representation  of  the  face  of  the  sun.      That  this  is  really  the  case,  is  evidently  seen  by  making  the  expe- 
riment in  the  time  of  a  solar  eclipse,  when  the    image  on  the  screen,  instead  of  appearing  round,  appears 
horned,  like  the  sun  itself.*     In  like  manner,  if  a  pin-hole  in  a  card  be  held  between  a  candle  and  a  piece 
of  white  paper  in  a  dark  room,  an  exact  representation  of  the  flame,  but  inverted,  will  be  seen  depicted  on 
the  paper,  which  enlarges  as  the    paper  recedes  from    the  hole ;    and  if  in  a  dark  room  a  white  screen  be 
extended  at  a  few  feet  from    a   small    round   hole,  an  exact  picture  of  all  external  objects,  of  their  natural 
colours  and  forms,  will  be  seen  traced  upon  the  screen ;    moving  objects  being  represented  in    motion,  and 

Fig.  6.  quiescent  ones  at  rest.  (See  fig.  6.)  To  understand  this,  we  must  recollect  that  all  objects  exposed  to  light 
are  luminous  ;  that  from  every  physical  point  of  them  light  radiates  in  all  directions,  so  that  every  point  in 
the  screen  is  receiving  light  at  once  from  every  point  in  the  object.  The  same  may  be  said  of  the  hole  ; 
but  the  light  that  falls  on  the  hole  passes  through  it,  and  continues  its  course  in  straight  lines  behind. 
Thus  the  hole  becomes  the  vertex  of  a  conoidal  solid  prolonged  both  ways,  having  the  object  for  its 
base  at  one  end,  and  the  screen  at  the  other.  The  section  of  this  solid  by  the  screen  is  the  picture  we  see 
projected  on  it,  which  must  manifestly  be  exactly  similar  to  the  object,  and  inverted,  according  to  the  simplest 
rules  of  Geometry. 

Now  if  in  our  screen  receiving  (suppose)  the  image  of  the  sun  we  make  another  small  hole,  and  behind 
it  place  another  screen,  the  light  falling  on  the  space  occupied  by  this  hole  will  pass  beyond  it,  and  reach 
the  other  screen ;  but  it  is  clear  that  it  will  no  longer  dilate  itself,  after  passing  through  the  second  hole, 
and  form  another  image  of  the  whole  sun,  but  only  an  image  of  that  very  minute  portion  of  the  sun  which 
corresponds  to  the  space  occupied  in  his  image  on  the  first  screen  by  the  hole  made  there.  The  lines 
bounding  the  conoidal  surface  will  in  this  case  have  much  less  divergency,  and,  if  the  holes  be  small 
enough,  and  very  distant  from  each  other,  will  approach  to  physical  lines,  and  that  the  nearer,  as  the  holes 

Fig.  7.  are  smaller  and  their  distance  greater.  (See  fig.  7.)  If  we  conceive  the  holes  reduced  to  mere  physical  points, 
these  lines  form  what  we  call  rays  of  light.  Mathematically  speaking,  a  ray  of  light  is  an  infinitesimal 
pyramid,  having  for  its  vertex  a  luminous  point,  and  for  its  base  an  infinitely  small  portion  of  any  surface 
illuminated  by  it,  and  supposed  to  be  filled  with  the  luminous  emanation,  whatever  that  may  be.  This 
pyramid,  in  homogeneous  media,  and  when  the  course  of  the  ray  is  not  interrupted,  has,  as  we  have  seen, 


•  In  the  eclipse  of  September  7,  1820,  this  horned  appearance  was  very  striking  in  the  luminous  interstices  between  the  shadows  of 
small  irregular  objects,  as  the  leaves  of  trees.  Sec.     It  was  noticed  by  those  who  had  no  idea  of  its  cause. 


LIGHT.  343 

Light      its  sides  straight  lines.     If  cases  should  occur   (as  they  will)  when  the  course  of  the  light  is  curved,  or  sud-      Part  I. 
x—-\'-— ^  denly  broken,  we  may  still  conceive  such  a   pyramid  having  curved  or  broken  sides  to    correspond  ;    or  we  *—•->,—• 
may  (for  brevity's  sake)  substitute  for  it  a  mere  mathematical    line,  straight,  curved,  or  broken,  as  the  case 
may  be. 

9.  Light  requires  time  for  its  propagation.      Two    spectators    at    different    distances  from  a   luminous  object 

Velocity  of  suddenly  disclosed,  will  not  begin  to  see  it  at  the  same  mathematical  instant  of  time.     The  nearer  will  see 

''I?*1''  it  sooner  than  the  more  remote  ;    in  the  same  way  as  two  persons  at  unequal    distances  from    a   gun   hear 

the  report  at  different  moments.     In  like  manner,  if  a  luminous  object  be  suddenly  extinguished,  a  spectator 

will  continue  to  see  it  for  a  certain  time  afterwards,  as  if  it  still  continued  luminous,  and  this  time  will  be 

greater  the  farther  he  is  from  it.     The  interval  in  question  is,  however,  so  excessively  small  in  such  distances 

as  occur  on  the  earth's  surface,  as  to  be  absolutely  insensible ;  but  in  the  immense  expanse  of  the  celestial 

regions  the  case  is  different.     The  eclipses  and  emersions  of  Jupiter's  satellites  become  visible  much  sooner 

(nearly  a  quarter  of  an  hour)  when  the  earth  is  at  its  least  distance  from  Jupiter  than  when  at  its  greatest. 

Light  then  takes  time  to  travel  over  space.     It  has  a  finite,  though    immense  velocity,  viz.  192500  miles  per 

second;    and  this  important  conclusion,  deduced    by  calculation  from  the  phenomenon   just   mentioned,  and 

which,  if  it  stood  unsupported,  might  startle  us  with  its  vastness,  and  incline  us  to  look  out  for  some  other 

Aberration    mode  of   explanation,  receives  full  confirmation   from  another  astronomical   phenomenon,    the    aberration    of 

of  light.       light,  which  (without  entering  into  any  close  examination  of   the  mode  in  which  vision  is  produced)  may  be 

explained  as  follows : 

10.  Let  a  ray  of  light  from  a  star  S,  at  such  a  distance  that  all  rays  from  it  maybe  regarded  as  parallel,  be 

received  on  a  small  screen  A,  having  an  extremely  minute  opening  A  in  its  centre ;  and  let  that  ray  which 
Fig  1.  passes  through  the  opening  be  received  at  any  distance  A  B,  on  a  screen  B  perpendicular  to  its  direction; 
and  let  B  be  the  point  on  which  it  falls,  the  whole  apparatus  being  supposed  at  rest.  If  then  we  join  the 
points  A,  B  by  an  imaginary  line,  that  line  will  be  the  direction  in  which  the  ray  has  really  travelled,  and 
will  indicate  to  us  the  direction  of  the  star ;  and  the  angle  between  that  line  and  any  fixed  direction  (that 
of  the  plumb-line,  for  instance)  will  determine  the  star's  place  as  referred  to  that  fixed  direction.  For  sim- 
plicity, we  will  suppose  this  angle  nothing,  or  the  star  directly  vertical ;  then  the  point  B  on  which  the  ray 
fells  will  be  precisely  that  marked  by  a  plumb-line  let  fall  from  A ;  and  the  direction  in  which  we  judge  the 
star  to  lie  will  coincide  precisely  with  the  direction  of  gravity.  Such  will  be  the  case,  supposing  the  earth, 
the  spectator,  and  the  whole  apparatus  at  rest ;  but  now  suppose  them  carried  along  in  a  horizontal  direction 
AC,  B  D,  with  a  uniform  and  equal  velocity,  of  whose  existence  they  will  therefore  be  perfectly  insensible,  and 
the  pumb-line  will  hang  steadily  as  before,  and  coincide  with  the  same  point  of  the  screen.  At  the  moment 
when  the  ray  S  A  from  the  star  passes  through  the  orifice  A,  let  A,  B  be  the  respective  places  of  the  orifice, 
and  the  point  on  the  screen  vertically  below  it.  When  the  ray  has  passed  the  orifice,  it  will  pursue  its  course 
in  the  same  straight  line  S  A  B  as  before,  independent  of  the  motion  of  the  apparatus,  and  in  some  certain 

S         distance  A  B  \ 

time    I  =  — ; — : =  t  I   will  reach  the  lower  screen.     But  in  this  time  the  aperture,  screens,  and 

\      velocity  of  light          / 

plumb-line  will  have  moved  away  through  a  space 

_  ,  /  earth's  velocity  \ 

A  a  =  B  6  {  =  t  x  velocity  of  motion  =  A  B  x  • — ; 

\  velocity  of  light/ 

At  the  instant,  then,  that  the  ray  impinges  on  the  lower  screen,  the  plumb-line  will  hang,  not  from  A  on  B,  but 
from  a  on  b  ;  and  a  being  the  real  orifice,  and  B  the  real  point  of  incidence  of  the  light  on  the  screen,  the 
spectator,  judging  only  from  these  facts,  will  necessarily  be  led  to  regard  the  ray  as  having  deviated  from  its 
vertical  direction,  and  as  inclining  from  the  vertical,  in  the  direction  of  the  earth's  motion  through  an  angle  whose 

A  a  earth's  velocity 

tangent  is  —  -• --  —  or  —. 

AB  velocity  of  light 

The  eye  is  such  an  apparatus.  Its  retina  is  the  screen  on  which  the  light  of  the  star  or  luminary  falls, 
and  we  judge  of  its  place  only  by  the  actual  point  on  this  screen  where  the  impression  is  made.  The 
pupil  is  the  orifice.  If,  the  eye  preserving  a  fixed  direction,  the  whole  body  be  carried  to  one  side  with  a 
velocity  commensurate  to  that  of  light,  before  the  rays  can  traverse  the  space  which  separates  the  pupil  from 
the  retina,  the  latter  will  have  shifted  its  place  ;  and  the  point  which  receives  the  impression  is  no  longer 
the  same  which  would  have  received  it  had  the  eye  and  spectator  remained  at  rest ;  and  this  deviation  is  the 
aberration  of  light. 

Every  spectator  on  the  earth  participates  in  the  general  motion  of  the  whole  earth,  which  in  its  annual 
orbit  about  the  sun  is  very  rapid,  and  though  far  from  equal  to  that  of  light,  is  by  no  means  insensible, 
compared  to  it.  Hence  the  stars,  the  sun,  and  planets,  all  appear  removed  from  their  true  places  in  the 
direction  in  which  the  earth  is  moving. 

13.  This  direction  is  varying  every  instant,  as  the  earth  describes  an  orbit  round  the  sun.     The  direction  therefore 

of  the  apparent  displacement  of  any  star  from  its  true  situation  continually  changes,  i.  e.  the  apparent  place 
describes  a  small  orbit  about  the  true.  This  phenomenon  is  that  alluded  to.  It  was  noticed  as  a  fact  by 
Bradley,  while  ignorant  of  its  cause,  that  the  stars  appear  to  describe  annually  small  ellipses  in  the  heavens 
of  about  40"  in  diameter.  The  discovery  of  the  velocity  of  light  by  the  eclipses  of  Jupiter's  satellites,  then 
recently  made  by  Roemer,  however,  soon  furnished  its  explanation.  Later  observations,  especially  those  of 
Brinkley  and  Struve,  have  enabled  us  to  assign,  with  great  precision,  the  numerical  amount  of  this  inequality, 
and  thence  to  deduce  the  velocity  of  light,  which  by  this  method  comes  out  191515  miles  per  second,  differing 


344  L  I  G  II  T. 

Light,      from  the  former  only  by  a  two  hundredth  part  of  its  whole  quantity.     This  determination  is  certainly  to  he     Part  I. 
v-^~v— ^  preferred.  ^-^ ~v*™ 

14.  But  this  is  not  the  only  information  respecting  light  which  astronomical  observations  furnish.      We  learn 
Light  uni-  from  them  also,  "  That  the  light  of  the  sun,  the  planets,  and  all  the  fixed  stars,  travels  with  one  and  the  same 
form   in  its  velocity."     Now  as  we  know  these  bodies  to  be  at  different  and  variable  distances  from  us,  we  hence  conclude 
motion.        tkat  tjje  veiocity  Of  light  ;s  independent  of   the  particular  source  from  which  it  emanates,   and  the  distance 

over  which  it  has  travelled  before  reaching  our  eye. 

15.  The  velocity  of  light,  therefore,  in  that  free  and  perhaps  void  space  which  intervenes  between  us  and  the 
planets  and  fixed  stars,  cannot  be  supposed  other  than  uniform  ;  and  the  calculations  of  the  eclipses  of  Jupiter's 
satellites,  and  the  places  of  the  distant  planets  made  on  this  supposition  agreeing  with  observation,  prove  it  to 
be  so.     In  entering  such  media  as  it  traverses,  when  arrived  within  the  limits  of  the  atmospheres  of  the  earth 
and  other  planets,  we  shall  find  reason  hereafter  to  conclude  that  its   velocity  undergoes  a  change  ;    but,  at 
all  events,   we  have    no  reason  to  suppose  it  to  differ  in  different  parts  of  one  and  the  same  homogeneous 
medium. 

16.  The  enormous  velocity  here  assigned  to  light,  surprising  as  it  may  seem,  is  among  those  conclusions  which 
Velocity  of  rest  on  the  best  evidence  that  science  can  afford,  and  may  serve  to  prepare  us  for  other  yet  more    amazing 
light  illus-    numerical  estimates.     It  is  when  we  attempt  to  measure  the  vastness  of  the  phenomena  of   nature  with   our 

'  .y     feeble  scale  of  units,  such  as  we  are  conversant  with  on  this  our  planet,  that  we  become  sensible  of  its  insig- 
solls  nificance  in  the  system  of  the  universe.     Demonstrably  true  as  are  the  results,  they  fail  to  give  us  distinct  con- 

ceptions ;  we  are  lost  in  the  immensity  of  our  numbers,  and  must  have  recourse  to  other  ways  of  rendering 
them  sensible.  A  cannon  ball  would  require  seventeen  years  at  least  to  reach  the  sun,  supposing  its  velocity 
to  continue  uniform  from  the  moment  of  its  discharge.  Yet  light  travels  over  the  same  space  in  7^  minutes. 
The  swiftest  bird,  at  its  utmost  speed,  would  require  nearly  three  weeks  to  make  the  tour  of  the  earth.  Light 
performs  the  same  distance  in  much  less  time  than  is  required  for  a  single  stroke  of  his  wing ;  yet  its  rapidity 
is  but  commensurate  to  the  distances  it  has  to  travel.  It  is  demonstrable  that  light  cannot  possibly  arrive  at 
our  system  from  the  nearest  of  the  fixed  stars  in  less  than  five  years,  and  telescopes  disclose  to  us  objects 
probably  many  thousand  times  more  remote. 

But  these  are  considerations  which   belong  rather  to  astronomy  than  to  the   present  subject ;  and  we  will, 
therefore,  return  to  the  consideration  of  the  phenomena  of  emitted  light. 

§  II.     Of  Photometry. 

Of  these,  one  of  the  most  striking  is  certainly  the  diminution  of  the  illuminating  power  oC  any  source  of  light, 
ni's'hes  asthe  arising  from  an  increase  of  its  distance.  We  see  very  well  to  read  by  the  light  of  a  candle  at  a  certain  distance : 
distance  of  remove  the  candle  twice,  or  ten  times  as  far,  and  we  can  see  to  read  no  longer. 

its  source         The  numerical  estimation  of  the  degrees  of  intensity  of  light  constitutes  that  branch  of  optics  which  is  termed 
increases.      Photometry.    (0u>«,  fierpw.) 

If  light  be   a  material  emanation,  a  something  scattered  in  minute   particles  in   all  directions,  it  is  obvious 
invereeb'al that  tlle  same  quantitv  which  is  diffused  over  the  surface  of  a  sphere  concentric  with  the  luminous  points,  if  it 
the  squra   continue  its  course,  will  successively  be  diffused  over  larger  and  larger  concentric  spherical  surfaces ;  and  that  its 
of  the         intensity,  or  the  number  of  rays  which  fall  on  a  given  space,  in  each  will  be  inversely  as  the  whole  surfaces  over 
distance.      which  it  is  diffused ;  that  is,  inversely  as  the  squares  of  their  radii,  or  of  their  distances  from  the  source  of  light. 
Without  assuming  this  hypothesis,  the  same  thing  may  be  rendered  evident  as  follows.     Let  a  candle  be  placed 
behind  an  opaque  screen  full  of  small  equal  and  similar  holes ;   the  light  will  shine  through  these,  and  be  inter- 
cepted in  all  other  parts,  forming  a  pyramidal  bundle  of  rays,  having  the  candle  in  the  common  vertex.     If  a 
sheet  of  white  paper  be  placed  behind  this,  it  will   be  seen  dotted  over  with  small  luminous  specks,  disposed 
exactly  as   the  holes  in  the  screen.     Suppose  the  holes  so  small,  their  number  so  great,  and  the  eye  so  distant 
from  the  paper  that  it  cannot  distinguish  the  individual  specks,  it  will  still  receive  a  general  impression  of  bright- 
ness ;    the  paper  will  appear  illuminated,  and  present  a  mottled  appearance,  which,  however,  will   grow   more 
uniform   as   the  holes  are  smaller,  and  closer,  and   the   eye  more  distant ;  and  if  extremely  so,  the  paper  will 
appear  uniformly  bright.     Now,  if  every  alternate  hole  be  stopped,  the  paper  will  manifestly  receive  only  half 
the  light,  and  will  therefore  be  only  half  as  much  illuminated,  and  ceeteris  paribus,  the  degree  of  illumination 
is  proportional  to  the  number  of  the  holes  in  the  screen,  or  to  the  number  of  equally  illuminated  specks  on 
its  surface,   i.  e.  if  the  specks  be  infinitely  diminished  in    size,  and   infinitely    increased   in    number,  to    the 
number  of  rays  which  fall  on  it  from  the  original  source  of  light. 

19  Let  a  screen,  so  pierced  with  innumerable  equal  and  very  small  holes  in  the  manner  described,  be  placed 

at  a  given  distance  (I  yard)  from  a  candle;  and  in  the  diverging  pyramid  of  rays  behind  it  place  a  small 
piece  of  white  paper  of  a  given  area,  (I  square  inch,  for  instance,)  so  as  to  be  entirely  included  in  the  pyramid. 
It  is  manifest  that  the  number  of  rays  which  fall  on  it  will  be  fewer  as  it  is  placed  farther  from  the  screen. 
because  the  whole  number  which  pass  the  screen  are  scattered  continually  over  a  larger  and  larger  space. 
Thus  were  it  close  to  the  screen  it  would  receive  a  number  equal  to  that  of  the  holes  in  a  square  inch  of  the 
screen,  but  at  twice  the  distance  (2  yards)  from  the  candle  this  number  will  be  spread  over  four  square  inches 
by  their  divergence,  and  the  paper  can  therefore  receive  only  a  fourth  part  of  that  number.  If,  therefore, 

its  illumination  when  close  to  the  screen  be  represented  by  I,   it  will  at  twice  the   distance   be  only—,  and 


LIGHT.  345 

Light.  I  Part  I- 

i_r-    -m^1  •*  D  times  the  assumed  unit  of  distance,  its  illumination  will  be  —  ,  the    areas    of  sections  of  a   pyramid 

being  as  the  squares  of  their  distances  from  the  vertex. 

20.  As  this  reasoning  is  independent  of  the  number  and  size  of  the  holes,  and  therefore  of  the  ratio  of  the 
sum  of  their  areas  to  that  of  the  unperforated  part  of  the  screen,  we  may  conceive  this  ratio  increased  ad 
infinitum.     The  screen  then  disappears,  and  the  paper  is  freely  illuminated.     Hence  we  conclude  that  when 
a  small  plane  object  of  given  area  is  freely  and  perpendicularly  exposed  to  a  luminary  at  different  distances, 
the  quantity  of  light  it  receives,  or  the  degree  of  its  illumination,  is  inversely  as  the  squares  of  its  distance 
from  the  luminary,  cteteris  paribus. 

21.  If  a  single  candle  be  placed  before  a  system  of  holes  in  a  screen,  as  before,  and  the  rays  received  on  a 
Illumination  screen  at  a  given  distance,  (1,)  the  degree  of  illumination  may  be  represented  by  a  given  quantity,  I.     Now 
proportional  if  a  second  candle  be  placed  immediately  behind  the  other,  and  close  to  it,  so  as  to  shine  through  )he  same 

i  the  num-  holes,  the  illumination  of  the  screen  is  perceived  to  be  increased,  though  the  number  and  size  of  the  illu- 

t-lisity  of""  wi'nated  points  on  it  be  the  same.     Each  point  is  then  said  to  be    more    intensely  illuminated.     Now,  (the 

the  ravsj      eye  being  all  along  supposed  so  distant,  and  the  illuminated  points  so  small  as  to  produce  only  a  general 

sense  of  brightness,  without  distinguishing  the  individual  points,)  if  the  one  candle  be  shifted  a  little  sideways, 

without  altering  its  distance,  the  illumination  of  the  paper  will  not  be  altered.      In  this  ease  the  number  of 

illuminated  points  is  doubled,  but  each  is  illuminated  with  only  half  the  light  it  had  before.     The  same  holds 

for  any  number  of  candles.     Hence  we  conclude  that  the  illumination    of    a  surface    is    constant  when  the 

number  of  rays  it  receives  is  inversely  as  the  intensity  of  each,  and  that  consequently  the  degree  of  illumination 

is  proportional  to  the  number  and  intensity  of  the  rays  jointly. 

22.  Now  if  for  any  number  of  candles  placed  side  by  side  we  substitute  mere  physical  luminous  points,  each 
and  to   the  of  these  will  be  the  vertex  of  a  pyramid  of  rays,  and  the  number  of  equally  illuminated  points  in  the  paper, 
area  of  the.  and  therefore  illuminations  will  be  proportional  to  the  number  of  such  points.     If  we  conceive  the  number 

"*7  of  these  increased,  and  their  size  diminished  ad  iiifinitum,  so  as  to  form  a  continuous  luminous  surface,  their 
number  will  be  represented  by  its  area.  Hence  the  illumination  of  the  paper  will  be,  ceeteris  paribus,  as  the 
area  of  the  illuminating  surface,  (supposed  of  uniform  brightness.) 

23.  Uniting  all  these  circumstances,  we  see  that  when  a  given  object  is  enlightened  by  a  luminous  surface  of 
General  ex-  small  but  sensible  size,  the  degree  of  its  illumination  is  proportional  to  the 

area  of  the  luminous  surface  X  intensity  of  its  illuminating  power 
square  of  the  distance  of  the  surface  illuminated. 

21.  The  foregoing  reasoning  applies  only  to  the  case  when  the  luminous  disc  is  a  small  portion  of  a  spherical 

Oblique  il-  surface  concentric  with  the  illuminated  object,  in  which  case  all  its  points  are  equidistant  from  it,  and  all  the 

lumination.   light  falls  perpendicularly  on  the  object.     When  the  object  is  exposed  obliquely,  conceive  its  surface  divided 

into  equal  infinitely  small  portions,  and  regard  each  of  them   as  the  base  of   an  oblique  pyramid,  having  its 

vertex  at  any  one  point  of  the  luminary  ;     then  will  the  perpendicular  section   of   this  pyramid  at  the  same 

distance  be  equal  to  the  base  x  sine  of   inclination  of  the  base  to  the   axis,  or  the  element  of  the  illuminated 

surface  X  by  the  sine  of  the  inclination  of  the  ray.     But  the  number  of  rays  which  falls  on  the  base  is  evidently 

equal  to  those  which  fall  on  the  section,  and  being  spread  over  a  larger  area  their  effect  will  be  to  illuminate  it 

less  intensely  in  the  proportion  of  the  area  of  the  section  to  that  of  the  base,  i.  e.  in  the  proportion  of  the  sine  of 

inclination  to  radius.     But  the   illumination  of  the  section  is  equal   to  the 

area  of  the  luminary  x  intrinsic  brightness 
(distance)  * 

therefore  that  of  the  elementary  surface  equals   this  fraction  multiplied  by  the  sine  of  the  rays'  inclination  ; 
or,   calling   A   the   area   of   the    luminary,    I    its   intrinsic    brightness,   D   its  distance,  and   0  the  inclination 

of  the  ray  to  the  illuminated  surface   —  '—^  --  will  represent  the  intensity  of  illumination 

25.  If  L  represent  the  absolute  quantity  of  light  emitted  by  the  luminary  in  a  given  direction,  which  may  be  called 
its  absolute  light,  we  have  L  =  A  X  I,  provided  the   surface  of    the  luminary  be  perpendicular    to   the   given 
direction.     If  not,  A  must  represent  the  area  of  the  section  of  a  cylindroidal  surface  bounded  by  the  outline  of 

the  luminary,  and  having  its  axis  parallel  to  the  given  direction  ;   consequently    -  —  represents  in  this  case 

the  intensity  of  illumination  of  the  elementary  surface. 

To  illustrate  the  application  of  these  principles  we  will  resolve  the  following 

PROULEM. 

26.  A  small  white  surface  is  laid  horizontally  on  a  table,  and  illuminated  by  a  candle  placed  at  a  given  (hori- 
zontal) distance  :  What  ought  to  be  the  height  of  the  flame,  so  as  to  give  the  greatest  possible  illumination  to  the 
surface  ? 

Let  A  be  the  surface,  B  C  the   candle.      Put  AB=a.AC  =  D;    B  C  =   -/D2  —  a4.      Then,  since  the 


sin  C  A  B  C  B  —  a 

illumination  of  A  is,  ceelens  paribus,  as  —  7~F?-  •  or  as  -.  (,3-   =  --  ~-^  --  (=  F)  we  have  to  make  this 

VOL.  IV.  2  Z 


346  LIGHT. 

Light,     quantity  a  maximum ;  consequently  d  F  =  o,  or  d ,  F 4  =  o,  that  is, 


or  D  =  a  .  V   '  -    and   B  c  =  A/U2-  «*  =  -     =  =  0.707  x  A  B. 
8  */  2 

27.  Definition.     The  apparent  superficial  magnitude,  or  the  apparent  magnitude  of  any  object,  is  a  portion  of  a 
Apparent  spherical  surface  described  about  the  eye  as  a  centre,  with  a  radius  equal  to   1,  and  bounded  by  an  outline 
magnitude  being  the  intersection  of  this  spherical  surface  with  a  conoidal  surface,  having  the  object  for  its  base  and  the 
defined.  eye  jor  j,s  vcrtex. 

28.  Hence  the  apparent  superficial  magnitude  of  a  small  object  is  directly  as  the  area  of  a  section  (perpendi- 
cular to  the  line  of  sight)  of  this  conoidal  surface,  at  the  place  of  the  object,  and  inversely  as  the  square  of  its 
distance.     If  the  object  be  a  surface  perpendicular  to  the  line  of  sight,  this  ratio  reduces  itself  to  the  area 
of  the  object  divided  by  the  square  of  its  distance. 

29.  Definition.  The  real  intrinsic  brightness  of  a   luminous  object  is  the  intensity  of  the  light  of  each  physical 
Real  intrin-  point  in  its  surface,  or  the    numeric.il  measure  of    the  degree  in  which  such  a   point  (of  given  magnitude) 
sic  bright-    would  illuminate  a  given  object  at  a    given  distance,  referred  to  some  standard  degree  of  illumination  as  a 
ness  defined.  unjt      When  we  speak  simply  of  intrinsic  brightness,  real  intrinsic  brightness  is  meant. 

30.  Carol.  1.  Consequently  the  degree  of  illumination  of  an  object  exposed  perpendicularly  to  a  luminary  is  as 
the  apparent  magnitude  of  the  luminary  and  its  intrinsic  brightness  jointly. 

31.  Carol.  2.   Conversely,  if  these  remain  the  same,  the  degree  of  illumination  remains  the  same.     For  example, 
the  illumination  of  direct  sunshine  is  the  same  as  would  be  produced  by  a  circular    portion  of  the  surface 
of  the  sun  of  one  inch  in  diameter,  placed  at  about  10  feet  from  the  illuminated  object,  and  the  rest  of  the  sun 
annihilated ;  for  such  a  circular  portion  would  have  the  same  apparent  superficial  magnitude  as  the  sun  itself 
This  will  serve  to  give  some  idea  of  the  intense  brightness  of  the  sun's  disc. 

32.  Definition.     The  apparent  intrinsic  brightness  of  any  object,  or  luminary,  is  the  degree  of  illumination  of 
Apparent  its  image  or  picture  at  the  bottom  of  the  eye.     It  is  this  illumination  only  by  which  we  judge  of  brightness, 
intrinsic  A.  luminary  may  in  reality  be  ever  so  much  brighter  than  another ;  but  if  by  any  cause  the  illumination  of  its 
brightness.  {mag.e  m  the  eye  be  enfeebled,  it  will  appear  no  brighter  than  in  proportion  to  its  diminished  intensity.     Thus 

we  can  gaze  steadily  at  the  sun  through  a  dark  glass,  or  the  vapours  of  the  horizon. 

Definition.  The  absolute  light  of  a  luminary  is  the  sum  of  the  areas  of  its  elementary  portions,  each  multi- 
Absolute  plied  by  its  own  intrinsic  brightness ;  or,  if  every  part  of  the  surface  be  equally  bright,  simply  the  area  multi- 
defined  ph'ed  by  the  intrinsic  brightness.  It  is,  therefore,  the  same  quantity  as  that  above  represented  by  L. 

'34.  Definition.    The  apparent  light  of  an  object  is  the  total  quantity  of  light  which  enters  our  eyes  from  it, 

Apparent     however  distributed  on  the  retina. 

light  In  common  language,  when  we  speak  of  the  brightness  of  an  object  of  considerable  size,  we  often  mean  its 

defined.        apparent  intrinsic  brightness.     When,  however,  the  object  has  no  sensible  size,  as  a  star,  we  always  mean  its 

35-       apparent  light,  (or,  as  it   might  be  termed,  its  apparent  absolute  brightness,)  because,  as  we  cannot  distinguish 

such  an  object  into  parts,  we  can  only  be  affected  by  its  total  light  indiscriminately.     The  same  holds  good  with 

all  small  objects  which  require  attention  to  distinguish  them  into  parts.     Optical  writers  have  occasionally  fallen 

into  much  confusion  for  want  of  attending  to  these  distinctions. 

36".  As  we  recede  from  a  luminary,  its  apparent  light  diminishes,  from  two  causes  ;  first,  our  eyes,  being  of  a  given 

111  size,  present  a  given  area  to  its  light,  and  therefore  receive  from  it  a  quantity  of  light  inversely  as  the  square  of 

"iflu'b'v™"  *^e  (i'stance  i  secondly,  in  passing  through  the  air,  a  portion  of  the  light  is   stopped,  and  lost  from  its  want  of 

instance.      perfect  transparency.     This,  however,  we  will  not  now  consider.     In  virtue  of    the  first  cause  only,  then,  the 

apparent  light  of  a  luminary  is  inversely  as  the  square  of  its  distance,  and  directly  as  its  absolute  light, 
37.  The  apparent  intrinsic  brightness  is  equal  to  the  apparent  light  divided  by  the  area  of  the  picture  on  the  retina 

Objects  ap-  of  our  eye.     But  this  area  is  as  the  apparent  superficial  magnitude  of  the  luminary,  that  is,  as  its  real  area  A 
pear  equally  » 

bright  at  all  divided  by  the  square  of  its  distance  D,  or  as  — — .      Moreover,  the   apparent  light,  as  we  have   just  seen,  is  as 
distances.  Da 

A  I 

• where  I  is  the  real  intrinsic  brightness.     Consequently  the  apparent  intrinsic  brightness  is  proportional  to 

A I  A 

H-   ,  or  simply  to  I,  and  is  independent  on  A  or  D.     The  apparent  intrinsic  brightness  is,  therefore, 

the  same  at  all  distances,  and  is  simply  proportional  to  the  real  intrinsic  brightness  of  the  object.  This  con- 
In  what  elusion  is  usually  announced  by  optical  writers  by  saying,  that  objects  appear  equally  bright  at  all  distances, 
sense  to  be  which  must  be  understood  only  of  apparent  intrinsic  brightness,  and  the  truth  of  which  supposes  also  that  no  loss 
understood,  of  li^ht  takes  place  in  the  media  traversed. 

38.  The  anyle  of  emanation  of  a  ray  of  light  from  a  luminous  surface  is  the  inclination  of  the  ray  to  the  surface  at 

Angle  of      the  p0i,,t  from  which  it  emanates. 

A  question  has  been  agitated  among  optical  philosophers,  whether  the  intensity  of  the  light  of  luminous  bodies 

au        be   the  same  in  all  directions  ;  or  whether,  on  the  other  hand,  it  be  not  dependent  on  the  angle  of  emanation. 

Euler,  in  his  Reflexions  sur  les  divers  degres  de  la  lumiere  du  Soliel,  Hfc.  Berlin,  Mem.  1750,  p.  280,  has  adopted 


LIGHT.  347 

Light,     the  former  principle.     Lambert,  on  the  other  hand,  Photometria,  p.  41,  contends  that  the  intensity  of  the  light,     P«t  I. 

v— - •^~~- '  or  density  of  the  rays,  issuing1  from  a  luminous  surface  in  any  direction  is    proportional  to  the  line  of  the     ~"v" • 

Question      angle  of  emanation.     If  we  knew  the  intimate  nature  of  light,  and  the  real  mechanism  by  which  bodies  emit 

among  opti-  an(j  reflect  ;t>  jt  mijrat  be  possible  to  decide  this  question   a  priori.      If,  for  instance,  we  were  assured  that 

pecUno'the  %nt   emanated   strictly  and  solely  from  the  molecules  situated  on  the  external  surface  of  bodies,  and  that  the 

dependence  emanation  from  each  physical  point  of  the  surface  were  totally  uninfluenced  by  the  rest  of  the  molecules  of 

of  the  emis- which  the  body  consists,  and  dispersed  itself  equally  in  all  directions,  then,  since  every  point  of  a  plane  surface 

sion  of  light  js  visible  t0  an  eve  wherever  situated  above  it,  and  each  is  supposed  to  send  the  same  number  of  rays  to  the 

'    eye  in  an  oblique  as  in  a  perpendicular  situation,  the  total  light  received  from  a  given  area  of  the  surface  in 

tjon  the  eye  ought  to  be  the  same  at  all  angles  of  emanation.     But  as  the  apparent  magnitude  of  this  area  is  as 

the  sine  of  its  inclination  to  the  line  of  sight,  i.  e.  of  its  angle  of  emanation,  this  light  is  distributed  over  a  less 

apparent  area ;  and  therefore  its  intensity,  or  the  apparent  brightness  of  the  surface,  should  be  increased  in  the 

inverse  ratio  of  the  sine  of  the  angle  of  emanation.     On  the  other  hand,  if,  as  there  is  every  reason  to  suppose, 

light   emanates,  not  strictly  from  the  surfaces  of  bodies,  but  from   sensible  depths  within  their  substance ;  if  the 

surfaces  themselves  be  not  true  mathematical  planes,  but  consist  of  a  series  of  physical  points  retained  in  their 

places  by  attractive  and  repulsive  forces,  and  if  the  intensity  of  emanation  of  each  of  these  points  depend  in  any 

way  on  its  relation  to  the  points  adjacent,  there    is    no    reason,  a  priori,  to  suppose  the  equal  emanation  <.,i 

light  in  all  directions ;    and  to  find  what  its  law  really  is,  we  must  have  recourse  to  direct  observation. 

Astronomy  teaches  us  that  the  sun  is  a  sphere.  Hence  the  several  par's  of  its  visible  disc  appear  to  us 
under  every  possible  angle  of  inclination.  Now  if  we  examine  the  surface  of  the  sun  with  a  telescope,  the 
circumference  certainly  does  not  appear  brighter  than  the  centre.  But  if  the  hypothesis  of  equal  emanation  were 
correct,  the  brightness  ought  to  increase  from  the  centre  outwards,  and  should  become  infinite  at  the  edges,  so 
that  the  disc  ought  to  appear  surrounded  by  an  annulus  of  infinitely  greater  splendour  than  the  central  parts.  To 
this  it  may,  however,  be  justly  objected,  that  as  the  surface  of  the  sun  is  obviously  though  generally  spherical, 
yet  full  of  local  irregularities,  every  minute  portion  of  it  may  be  regarded  as  presenting  every  possible  variety 
of  inclination  to  our  eye  ;  and  the  brightness  of  every  part  being  thus  an  average  of  all  the  gradations  of  which 
it  is  susceptible,  should  be  alike  throughout. 

40  Bouguer,  in  his  Traite"  d'Optique,  Paris,  1760,  p.  90,  states  himself  to  have  found,  by  direct  comparison,  that 
the  central  portions  of  the  disc  of  the  sun  are  really  much  more  luminous  than  the  borders.     A  result  so  extra- 
ordinary, however,  and  so  apparently  incompatible  with  all  we  know  of  the  constitution  of  the  sun  and  the  mode 
of  emission  of  light  from  its  surface,  would  require  to  be  verified  by  very  careful  and  delicate  reexamination.     If 
found  correct,  the  only  way  of  accounting  for  it  would  be  to  suppose  a  dense  and  imperfectly  transparent 
atmosphere  of  great  extent  floating  above  the  luminous  clouds  which  form  its  visible  surface.     This  is  certainly 
possible,  but  our  ignorance  on  the  subject  renders  it  unphilosophical  to  resort  to  a  body  so  little  within  our  reach 
for  the  establishment  of  any  fundamental  law  of  emanation.     The  objection  above  advanced,  it  will  be  observed, 
applies  with  nearly  the  same  force  to  all  surfaces.     If  we  examine  a  piece  of  white  paper  with  a  magnifier, 
we  shall  find  its  texture  to  be  in  the  last  degree  rough  and  coarse,  presenting  no  approach  to  a  plane;  and 
so  of  all  surfaces  rough  enough  to  reflect  light  in  all  directions. 

41  However,  as  it  is  only  with  such  luminous  surfaces  as  occur  in  nature  that  we  have  any  concern,  we  must 
Surfaces  take  their  properties  as  we  find  them  ;  and,  waiving  all  consideration  of  what  would  be  the  law  of  emanation 
appear  from  a  mathematical  surface,  it  may  be  stated  as  a  result  of  observation,  that  luminous  surfaces  appear  equally 
equally  bright  at  all  angles  of  inclination  to  the  line  of  sight. 

angles  *'          ^"n's  may  ^e  tr'et*  w'^  a  surface  °f  red-hot  iron ;  its  apparent  intrinsic  brightness  is  not  sensibly  increased 
by  inclining  it  obliquely  to  the  eye. 

42.  If  we  take  a  smooth  square  bar  of  iron,  or  better,  of  silver,  or  a  polished  cylinder  of  either  metal,  heated 
Experimen-  to  redness,  into  a  dark  room,  the  cylinder  will  appear  equally  bright  in  the  middle  of  its  convexity  next  the 
tal  proof  of  eye,  and  at  the  edges,  and  cannot  be  distinguished  at  all  from  a  flat  bar;    and  the  square  bar,  when  so  pre- 
law of   sente(j  as  (o  nave  two  of  its  sides  at  very  different  angles  to  the  line  of  sight,  will  still  appear  of  perfectly 

equable  brightness,  nor  can  the  angle  separating  the  adjacent  sides  be  at  all  discerned  ;  and  if  the  whole 
bar  be  turned  round  on  its  axis,  the  motion  can  only  be  recognised  by  an  alternating  increase  and  decrease 
of  its  apparent  diameter,  according  as  it  is  seen  alternately  diagonally  and  laterally,  its  appearance  being 
always  that  of  a  flat  plate  perpendicularly  exposed  to  the  eye.  These  and  similar  experiments  with  surfaces 
artificially  illuminated,  which  the  reader  will  have  no  difficulty  in  imagining  and  making,  as  well  as  those 
recorded  by  Mr.  Ritchie  in  the  Edinburgh  Philosophical  Journal,  are  sufficient  to  establish  the  principle 
announced  in  Article  42,  to  which  (for  the  reasons  already  mentioned)  the  observation  of  Bouguer  on  the 
unequal  brightness  of  the  sun's  disc  offers  no  conclusive  objection. 

43.  Hence  it  follows,  that  the  surfaces  of  luminous  bodies,  at  least  their  ultimate  molecules,  do  not  emit  light 
Law  of  the  with  equal  copiousness  in  all  directions ;  but  that,  on  the  contrary,  the  copiousness  of  emission,  in  any  direction, 
oblique        js  as  (fa  sjne  Of  ^e  ans,le  of  emanation  from  the  surface. 

emanation  J  J 

of  light. 

PROBLEM. 

44_  To  determine  the  intensity  of  illumination  of  a  small  plane  surface  any  how  exposed  to  the  rays  from   a 

luminary  of  any  given  size,  figure,  and  distance ;  the  luminary  being  supposed  uniformly  bright  in  every  part. 
Conceive  the  surface  of  the  luminary  divided  into  infinitesimal  elementary  portions,  of  which  let  each  be 
regarded  as  an  oblique  section  of  a  pyramid,  having  for  its  vertex  the  centre  of  the  infinitely  small  illuminated 

2z  2 


348  LIGHT. 

Light,      plane  B,  fig.  3.     Let  P  Q  be  any  such  portion,  and  let  the  pyramid  B  P  be  continued  till  it  meets  the  surface     Part  J, 
v*-'-y-7"'  of  the  heavens  in  p,  there  projecting'  the  surface  PQ  into  the  areola  pq,  and  let  the  whole  luminary  C  D  EF  <s 
I      mation  be  in  like  manner  projected  into  the  disc  cdef.     Let  n-Q  be  a  section  of  the  pyramid  APQ,  perpendicular 
by  any         to  '^s  ax's-     Then,  first,  the  plane  B  will  be  illuminated  by  the  element  P  Q,  just  as  it  would  be  by  a  surface 
luminary      "f  Q  equally  bright,  in  virtue  of  the  principle  just  established.     Hence   P  Q  is  equivalent  to  an  equally  bright 
investigated,  surface  tr  Q.     Again,  since  the  apparent  magnitude  of  irQ  seen  from  B  is  the  same  with  that  of  p  q,  the   area 
f'g-3-          57 Q  ;s  equivalent  to  an  equally  bright  area  pq  placed   at  p  i/,   (Art.  29,  30,  31,  Cor.  1,  2.)     P  Q  is,  therefore, 
equivalent  to  pq.     And  since  the  same  holds  good  of  every  other  elementary  portion  of  the  surface,  and  the  total 
light  received  by  B  is  the  sum  of  the   lights  it  receives  from  all  the  elements   of    the    luminary,  the  whole 
surface  CDEF  must  be  equivalent  to  its  projection  c  d  ef. 

45.  Hence  the  illumination  of  B  depends,  not  at  all  on  the  real,  but  only  on  the  apparent  figure  and  magni- 
tude of  the  luminary ;   and  whatever  the  luminary  be,  we  may  always  substitute  for  it  a  portion  of  the  visible 
heavens,  supposed  of  equal  intrinsic  brightness,  and  bounded  by  the  same  outline. 

46.  Thus,  instead  of  the  sun,  we  may  suppose  a  small  circle  equal  in  apparent  diameter  to  the  sun,  and  equally 
bright ;  instead  of   a  luminous  rectangle  perpendicular  to  the  illuminated  plane  B,  and  of  infinite  height,  as 
A  G  H  I,   fig.  3,  we    may  substitute    the    spherical    sector    Z  A  G,  bounded   by  the  two  vertical  circles    Z  A, 
Z  G,  and  so  on. 

47.  Let  then  p  q,  any  elementary  rectangle  infinitely  small  in  both  dimensions  of  the  spherical  surface,  be  repre- 
sented by  d4A,  so  that     /  /  d2  A  shall  represent  the  surface  cdef  itself;    then  if  we   put  z  =   the   zenith 

distance  Z  p  of  this  portion,  its  illuminating  power  on  A  will  be  d*  A  .  cos  z,  and  the  total  illuminating  power 
of  the  whole  surface  A  will  be 


=.//• 


A  .  cos  z. 


48.  Example  1.  To  find  the  illuminating  power  of  the  sector  ZAG  confined  between  any  two  vertical  circles 

Generalfbr-  and  the  horizon,  (fig.  3.)     Here,  putting  0  for  the  azimuth  of  the  element  d2  A,  if  we  consider  it  as  terminated 
mula  for  il-  jjy  two  contiguous  verticals  and  two  contiguous  parallels  of  altitude,  we  have  d2  A  =  dz  x  dO  .  sin  z.     Hence 

L  =    /  /  d9dz  .  sin  z  .  cos  z  =  £  /  /  dOdz  .  sin  2  z  =  %  I  (0  +  C)  d  z  .  sin  2  z  ; 


plane' 

and   extending   the   integral  from  0  =  o  to  0  =  A  G,  the   amplitude   of   the   sector,  (which  we  will    call  a,) 
we  get 

a       r  a 

L  =  -   Idz.  sin  2z  =  —  —  (C—  £cos2z) 

2  «/  2 

which  extended  from  Z  =  o  to  z  =  90°  becomes  simply  L  =  —  . 

49,  Carol.  1.  This  is  a  measure  of  the  illuminating  power  of  the  sector,  on  the  same  scale  that  that  of  an  infinitely 

small  area  (A)  placed  at  the  zenith  would  be  represented  by  A  itself.     Because  in  this  case 

cos  z  =  o,  and    /  /   d-  A.  cos  z  =  A. 

50  Carol.  2.   On  the  same  scale  the  illuminating  power  of  the  whole  hemisphere  is  ir  where  tr  =  3.14159535 

51.  Example  2.  Required  the  illuminating  power  of  a  circular  portion  of  the  heavens  whose  centre  is  the  zenith. 
Calling  z  the  zenith  distance  of  any  element,  and  0  its  azimuth,  we  shall  have,  as  before, 

d1  A  =  d  0  d  z  .  sin  z,  and  therefore  L  =  /  /  dOdz  .  sin  z  .  cos  z  ==  /  6  .  -  -  =  v  I  d  z  .  sin  2  z 

extending  the  integral  from  0  =  o  to  0  =  2  rr.     That  is  L  =  w  .  (const  —  £  cos  2  z)  which  being  made  to  vanish 
when  z  =  o  becomes  L  =  —-  (I  —  cos  2  z)  =  v  .  (sin  r)4 

52.  Carol.  3.  The  illuminating  power  of  a  circular  luminary,  whose  centre  is  in  the  zenith,  is  as  the  square  of 
the  sine  of  its  apparent  semidiameter. 

53.  Example  3.  Required  the  illuminating  power  of  any  circular  portion  of  the  heavens  whatever. 

tllumina-          Let  T  K  L  M  be  the  illuminating  circle  ;   conceive  it  composed  of  annuli  concentric  with   itself,  and  of  one 

ting  power  of  them,   X  Y  Z,   (fig.  4,)   let  X  J?  be  an  infinitesimal  parallelogram  terminated  by  contiguous  radii  S  X  and 

oi  any  cir-  r,        0  ,     .        .,  J 

mUr  nor,  »  x<  °  being  the  centre. 

t±ofP  an  Put  Z  S  =  a  ;  S  X  =  *,  Z  X  =  z, 

«1ul"y  Angle  Z  S  X  =  0,  S  T  =  r. 

bright 

Area  d2  A  =  XJT,  =  dx  x  d<f>.  sin  x 


.-.L  =  lld<pdx.sinj;.  cos  z. 
but,  by  spherical  trigonometry,         cos  z  =  cos  a  .  cos  *  .  +  sin  a  .  sin  x  cos  <p. 


LIGHT.  349 

l^B^t-  f*  f*  \  1  Part  I 

^-^— «^-  Therefore  L  =    /    I  dx  .  dfi.sin  x  I  cos  a .  cos  x  +  sin  a  .  sin  x  .  cos  0.  \ 

The  first  integration  performed  relative  to  0,  and  extended  from  0  =  o  to  0  =  360°,  or  2  TT,  gives 

L  =    /  d  x  .  sin  x  x  2  JT  .  cos  o  .  cos  i. 

After  which  integrating,  with  respect  to  x.  and  extending  the  integral  from  x  =  o  to  i  =  ST  =  r,  we  find 

L  =  -  —  (1  —  cos  2  r)  =  ir  .  cos  a  (sin  r)*. 

This  lesult  is  particularly  elegant  and  remarkable.  It  shows,  that  to  obtain  the  illuminating' effect  of  a  circular 
luminary  (of  any  apparent  diameter)  at  any  altitude,  on  a  horizontal  plane,  we  have  only  to  reduce  its 
illuminating  effect  when  in  the  zenith,  in  the  ratio  of  radius  to  the  cosine  of  the  zenith  distance,  or  sine 
of  the  altitude.  For  other  examples,  the  reader  may  consult  Lambert's  Photometria,  cap.  ii.  from  which  this 
is  taken. 

54.  If  the  illuminating  surface  be  not  equally  intrinsically  bright  in  every  part,  if  we  call  I  the  intrinsic  brightness 

General  ex-  of  the  element  d  *  A,  we  shall  have 
pression  for  (* f* 

illumination  L=     //    I  d  *  A  .  COS  3 

when  ttie  «/«/ 

luminary  is  for  the  general  formula  expressing  the  illuminating  power  of  the  surface  A.  The  moon,  Venus  and  Mercury  in 
bright"1"*  ^  tne'r  Pnases.  tne  sky  during  twilight,  a  white  sphere  illuminated  by  the  sun,  &c.  afford  examples  of  this  when 
throughout,  themselves  regarded  as  luminaries. 

PROBLEM. 

55_  To  compare  the  illumination  of  a  horizontal  plane  by  the  sun  in  the  zenith  with  the  illumination  it  would 

have  were  the  whole  surface  of  the  heavens  of  equal  brightness  with  the  sun. 

By  Art.  53  we  have  L  =  TT  .  cos  a  .  (sin  r)  2.  If,  therefore,  we  call  L  and  L'  the  two  illuminations  in 
question,  we  shall  have 

L  :  L'  :  :  77- .  cos  o° .  (sin  Q's  semidiam.)2  :  ir  .  cos  o° .  (sin  90°)* 

:  :  (sin  16')*  :   1  :  :   1   :  46166. 

56.  The  illumination  of  a  plane  in  contact  with  the  sun's  surface  is  the  same  as  that  of  a  plane  on  the  earth's 
Illumination  surface  illuminated  by  a  whole  hemisphere  of  equal  brightness  with  the  sun  in  the    zenith.     Hence  we   see 

s  that  the  illumination  of  such  a  plane  at  the  sun's  surface  would  be  nearly  50,000  times  greater  than  that  of 
the  earth's  surface  at  noon  under  the  equator.  Such  would  be  the  effect  (in  point  of  light  alone)  of 
bringing  the  earth's  surface  in  contact  with  the  sun's  \ 

57.  For  measuring  the  intensity  of  any  given  light,  various  instruments  called  Photometers  have  been  contrived, 
Photometers  many  of  which  have  little  to  recommend  them  on  the  score  of  exactness,  and  some  are  essentially  defective 

in  principle,  being  adapted  to  measure — not  the  illuminating — but  the  heating  power  of  the  rays  of   light  ; 
and,  therefore,  must  be  regarded  as  undeserving  the  name  of  photometers. 

58.  We  know  of  no  instrument,  no  contrivance,  as  yet,  by  which  light  alone  (as  such)  can  be  made  to  produce 
mechanical  motion,  so  as  to  mark  a  point  upon  a  scale,  or  in  any  way  to  give  a  direct  reading  off'  of  its 
intensity,  or  quantity,  at  any  moment.     This  obliges  us  to  refer  all  our  estimations  of  the  degrees  of  bright- 
ness at  once  to  our  organs  of  vision,  and  to  judge  of  their  amount  by  the  impression  they  produce  imme- 
diately on  our  sense  of  sight.     But  the  eye,  though  sensible  to  an  astonishing  range  of  different  degrees  of 
illumination,  is  (partly  on  that  very  account)  but  little  capable  of  judging  of  their  relative  strength,  or  even 

The  eye  au  of  recognising  their  identity  when  presented  at  intervals  of  time,  especially  at  distant  intervals.     In  this  manner 
B  ^      the  judgment  of  the  eye  is  as  little  to  be  depended  on  for  a  measure  of  light,  as  that  of  the  hand  would  be 
degrees  of   f°r  tne  weight  of  a  body  casually  presented.     This  uncertainty,  too,  is  increased  by  the  nature  of  the  organ 
illumination  itself,  which    is  in  a  constant  state  of  fluctuation  ;    the  opening  of  the  pupil,  which  admits  the  light,  being 
continually  expanding  and  contracting  by  the  stimulus  of  the    light    itself,  and  the  sensibility  of  the  nerves 
which  feel  the  impression  varying  at  every  instant.     Let  any  one  call  to  mind  the  blinding  and  overpower- 
ing effect  of  a  flash  of   lightning  in  a  dark  night,  compared  with  the  sensation  an  equally  vivid  flash  pro- 
duces in  full  daylight.      In  the  one  case  the   eye  is   painfully  affected,  and  the  violent  agitation  into  which 
the  nerves  of  the  retina  are  thrown  is  sensible  for  many  seconds  afterwards,  in  a  series  of  imaginary  alter- 
nations of  light  and  darkness.      By  day  no    such  effect  is  produced,  and  we  trace  the  course  of  the  flash, 
and  the  zig-zags  of  its  motion  with  perfect  distinctness  and  tranquillity,  and  without  any  of  those  ideas  of 
overpowering  intensity  which  previous  and  subsequent  total  darkness  attach  to  it. 

59.  But   yet  more.     When  two  unequally  illuminated  objects  (surfaces  of  white  paper,   for  instance)  are  pre- 
sented at  once  to  the  sight,  though  we  pronounce  immediately  on  the  existence  of  a  difference,  and  see  that 
one  is  brighter  than  the  other,  we  are  quite  unable  to  say  what  is  the  proportion  between  them.     Illuminate 
half  a  sheet  of  paper  by  the  light  of  one  candle,  and  the  other  half  by  that  of  several  ;    the  difference  will 
be  evident.     But  if  ten  different  persons  are  desired,  from  their  appearance  only,  to  guess  at  the  number  of 
candles  shining  on  each,  the  probability  is  that  no  two  will  agree.      Nay,  even  the  same  person  at  different 
times  will  form  different  judgments.     This  throws  additional  difficulty  in  the  way  of  photometrical  estimations, 
and  would  seem  to  render  this  one  of  the  most  delicate  and  difficult  departments  of  optics. 


350  LIGHT 

Light          However,  the  eye,  under  favourable  circumstances,  is  a  tolerably  exact  judge  of  the  equality  of  two  degrees     Part  I. 
^— ^V-^of  illumination  seen  at  once;  and  availing  ourselves  of  this,  we  may  by  proper   management  obtain   correct  •— —  /— • 

60.       information  as  to  the  relative  intensities  of  all  lights.     What  tiiese  favourable  circumstances  are,  we  come  now 
The  eye       to  consider. 

capabl  of  jgt  The  Degrees  of  illumination  compared  must  be  of  moderate  intensity.  If  so  bright  as  to  dazzle,  or 
iheSenua"ity  so  fa'nt  as  to  strain  the  eye,  no  correct  judgment  can  be  formed. 

of  two  de-  Hence,  it  is  rarely  adviseable  to  compare  two  luminaries  directly  with  each  other.  It  is  generally  better  to  let 
grees  of  il-  them  shine  on  a  smooth  white  surface,  and  judge  of  the  degree  in  which  they  illuminate  it ;  for  it  is  an  obvious 
lumination,  axjom>  That  two  luminaries  are  equal  in  absolute  light  when,  being  plan-d  at  equal  distances  from,  and  in  similar 
wiut""!)-"  situation*  with  respect  to,  a  given  smooth  white  surface,  or  two  equal  and  precisely  similar  white  surfaces,  they 
i  umstances.  illuminate  it  or  them  equally. 

Axiom  in  2nd.  The  luminaries,  or  illuminated  surfaces  compared,  must  be  of  equal  apparent  magnitude,  and  similar 
photometry,  figure,  and  of  such  small  dimensions  as  to  allow  of  the  illumination  in  every  part  of  each  being  regarded  as 

fi3-        the  same. 

64.  3rd.  They  must  be  brought  close  together,  in  apparent  contact ;   the  boundary  of  one  cutting  upon  that  of  the 
other  by  a  well-defined  straight  line. 

65.  4th.  They  should  be  viewed  at  once  by  the  same  eye,  and  not  one  by  one  eye,  and  the  other  by  the  other. 

66.  5th.    All  other  light  but.  that  of  the  two  objects  whose  illumination  is  compared  should  be  most  carefully 
excluded. 

67.  6th.  The  lights  which   illuminate   both    surfaces    must  be   of  the  same   colour.      Between   very    differently 
coloured  illuminations  no  exact  equalization  can  ever  be  obtained,  and  in  proportion  as  they  differ  our  judgment 
is  uncertain. 

68.  When  all  these  conditions  obtain,  we  can  pronounce  very  certainly  on  the  equality  or  inequality  of  two  illu- 
minations.    When   the  limit  between  them   cannot  be  perceived,  on  passing  the  eye  backwards  and  forwards 
across  it,  we  may  be  sure  that  their  lights  are  equal. 

69  Bouguer,  in  his  Traite  d'Optique,  1760,  p.  35,  has  applied  these  principles  to  the  measure  or  rather  the 

Boufuer's  comparison  of  different  degrees  of  illumination.  Two  surfaces  of  white  paper,  of  exactly  equal  size  and  re- 
principle  offlective  power,  (cut  from  the  same  piece  in  contact,)  are  illuminated,  the  one  by  the  light  whose  illuminating 
comparative  power  is  to  be  measured,  the  other  by  a  light  whose  intensity  can  be  varied  at  pleasure  by  an  increase  of 
photometry.  Distance,  an(j  can  therefore  be  exactly  estimated.  The  variable  light  is  to  be  removed,  or  approached,  till  the 

two  surfaces  are  judged  to  be  equally  bright,  when,  the  distances  of  the  luminaries  being  measured,  or  otherwise 

allowed  for,  the  measure  required  is  ascertained. 

70.  Mr.  Ritchie  has  lately  made  a  very  elegant  and  simple  application  of  this  principle.     His  photometer  consists 
Ritchie's      of  a  rectangular  box,   about  an  inch  and  a  half  or  two  inches   square,  open  at   both  ends,  of   which  A  B  C  I) 
photometer:  (fig  5)  ;g  a  section.      It  is  blackened  within,  to  absorb  extraneous  light.      Within,  inclined  at  angles  of  45°  to 

its  axis,  are  placed  two  rectangular  pieces  of  plane  looking-glass  FC,  FD,  cut  from  one  and  the  same  rectan- 
gular strip,  of  twice  the  length  of  either,  to  ensure  the  exact  equality  of  their  reflecting  powers,  and  fastened 
so  as  to  meet  at  F,  in  the  middle  of  a  narrow  slit  EFG  about  an  inch  long,  and  an  eighth  of  an  inch  broad, 
which  is  covered  with  a  slip  of  fine  tissue  or  oiled  paper.  The  rectangular  slit  should  have  a  slip  of  blackened 
card  at  F,  to  prevent  the  lights  reflected  from  the  looking-glasses  mingling  with  each  other. 

71.  Suppose  we  would  compare  the  illuminating  powers  of  two  sources  of  light  (two  flames,  for  instance)  Panel  Q. 
its  use.         They  must  be  placed  at  such  a  distance  from  each  other,  and  from  the  instrument  between  them,  that  the  lig-lit 

from  every  part  of  each  shall  fall  on  the  reflector  next  it,  and  be  reflected  to  the  corresponding  portion  of  the 
paper  E  F  or  F  G.  The  instrument  is  then  to  be  moved  nearer  to  the  one  or  the  other,  till  the  paper  on  either 
side  of  the  division  F  appears  equally  illuminated.  To  judge  of  this,  it  should  be  viewed  through  aprismoidal 
box  blackened  within,  one  end  resting  on  the  upper  part  A  B  of  the  photometer ;  the  other  applied  quite  close 
to  the  eye.  When  the  lights  are  thus  exactly  equalized,  it  is  clear  that  the  total  illuminating  powers  of  the 
luminaries  are  directly  as  the  squares  of  their  distances  from  the  middle  of  the  instrument. 

72.  By  means  of  this  instrument  we  are  furnished  with  an  easy  experimental  proof  of  the  decrease  of  light  as  the 
Experimen-  inverse  squares  of  the  distances.     For  if  we  place  four  candles  at  P,  and  one  at  Q,  (as  nearly  equal  as  possible, 
tal  proof  of  an(j  burning  with  equal  flames,)  it  is  found  that  the  portions  E  F,  GF  of  the  paper  will  be  equally  illuminated 

ii  o'f'Tht  when  the  distances  PF,  QF  are  as  2  :  1,  and  so  for  any  number  of  candles  at  each  side. 

aTthe   '  To  render  the  comparison  of  the  lights  more  exact,  the  equalization  of   the   lights   should  be  performed 

squares  of  several  times,  turning  the  instrument  end  for  end  each  time.  The  mean  of  the  several  determinations  will  then 
thedistances  be  very  near  the  truth. 

In  some  cases  the  looking-glasses  are  better  dispensed  with,  and  a  slip  of  paper  pasted  over  them,  so  as  to 
present  two  oblique  surfaces  of  white  paper  inclined  at  equal  angles  to  the  incident  light.  In  this  case  the 
paper  stretched  over  the  slit  EFG  is  taken  away,  and  the  white  surfaces  below  examined  and  compared.  One 
advantage  of  this  disposition  is  the  avoiding  of  a  black  interval  between  the  two  halves  of  the  slit,  which  renders 
the  exact  comparison  of  their  illuminations  somewhat  precarious. 

75.  If  the   lights  compared  be  of  different  colours  (as  daylight,  or   moonlight,  and   candlelight,)  their  precise 

Comparison  equalization  is  impracticable,  (art.  67.)  The  best  way  of  employing  the  instrument,  in  this  case,  is  to  move 
cit  k-hts  of  it  till  one  of  the  sides  of  the  slit  (in  spite  of  the  difference  of  colours)  is  judged  to  be  decidedly  the  brighter, 
different  am]  (|len  to  move  jt  the  other  way,  till  the  other  becomes  decidedly  the  brighter.  The  position  halfway  between 

these  points  is  to  be  taken  as  the  true  point  of  equal  illumination. 

75,  If  we  would  compare  the  degrees  of  illumination,  or  the  intrinsic  brightnesses  of  two  surfaces,  a  given  portion 

of  each  must  be  insulated  for  examination ;  this  may  be  best  done  by  the  adaptation  of  two  blackened  tubes  to 


LIGHT.  ;iol 

Light.      the  openings  of  the  photometer,  of  equal  length,  and  terminated  by  orifices  of  equal  area,  or  subtending  equal      fart  I. 
'ji-v    m'  angles  at  the  middle  of  the  instrument.     These,  of  course,  cut  off  equal   apparent  magnitudes  of  the  bright  >»^-y— •* 
Comparison  surfaceSi  the  light  of  which  is  then  to  be  equalized  on  the  oiled  papers   of  the  slit  E  F,   as  in  the  case  of 
bfbrightT    candles-  &c-  Bouguer,  Traite,  p.  31. 

nt><s  of  illu-      Another  method  of  comparing  the  intensity  of  the  light  from  two  luminaries,  which  is  also  very  ready  and 
minated        convenient,  and  possesses  in  some  cases  considerable  advantages,  has  been  proposed  by  Count  Rumford.    (See 
surfaces.       Phil.  Tram.,  vol.  84,  p.  67.)      It  consists   in  the  equalization  of  the  shadows  cast  by  them  on  a  white  surface 
^-        illuminated  by  them  both   at  once.     Suppose,  for  instance,  we  would   compare  the  illuminating  power  of  two 
^lg'  8'         flames  L  and  I  of  different  sizes,  or  from  different  combustibles,  as  of  wax  and  tallow.     Before  a  screen  C  D  of 
white  paper,  in  a  darkened  room,  place  a  blackened  cylindrical  stick  S,  and  let  the  flames  L  I  be  so  placed   as 
to  throw  the  shadows  A  B  of  the  stick  on  the  screen,  side  by  side,  and  with  an  interval  between  them  about 
equal  in  breadth  to  either  shadow.     Moreover  the  inclination  of  the  rays  L  S  A  and  I S  B  to  the  surface  of  the 
screen  must  be  adjusted  to  exact  equality.     The  brighter  flame  must  then  be  removed,  or  the  feebler  brought 
nearer  to  the  screen,  till  the  two  shadows  appear  of  equal  intensity,  when  their  distances  (or  the  distance  of  the 
screen)  from  the  lights  must  be  measured,  and  their  total   illuminating  powers  will   be  in  the  direct  ratio  of  the 
squares  of  the  distances.     The  rationale  of  this  is  obvious,  the  shadow  thrown  by  each  flame  is   illuminated  by 
the  light  of  the  other.     The  screen  by  the  sum  of  the  lights.     The  eye  in  this  case  judges  of  the  degrees  of  defal- 
cation of  brightness  from  this  sum ;  and  if  these  degrees  be  alike,  it  is  clear  that  the  remaining  illuminations  must 
be  equal. 

78.  This  method  becomes  uncertain  when  the  lights  are  of  considerable  size  and  near  the  screen,  as  the  penum- 
brae  of  the  shadows  prevent  any  fair  comparison  of  the  relative  intensities  of  their  central  portions.     It  is  still 
more  so,  and  can  hardly  be  used  when  the  lights  differ  considerably  in  colour.     Its  convenience,  however,  as  au 
extemporaneous  method,  requiring  no  apparatus  but  what  is  always  at  hand,  (as  the  use  of  a  blackened  stick, 
though  preferable,  is  not  essential,)  renders  it  often  useful  in  the  absence  of  more  refined  means. 

79.  It  may  happen  that  the  lights  to   be  compared   are  not  movable,  or  not  conveniently  so.     In  this  case  the 
When  the     equalization  of  the  shadows  may  be  performed  by  inclining  the  screen  at  different  angles  to  the  directions  in 
liglus  to  be  which  it  receives  the  light  of  each,  and  noting  the  angles  of  inclination  of  the  rays.     In  this  case  the  illumi- 
compared      natjnnr  powers  of  the  luminaries  are  as  the  squares  of  their  distances  directly,  and  the  sines  of  the  respective 
n-.ovable        angles  of  inclination  of  their  rays  to  the  screen  inversely. 

80.  When  a  ray  of  light  proceeds  in  empty  space,  or  in  a  perfectly  homogeneous  medium,  its  course,  as  we  have 
Modifica-      seen,  is  rectilinear,  and  its  velocity  uniform  ;    but  when  it  encounters  an  obstacle,  or  a   different  medium,  it 
tlonsof  light  undergoes  changes  or  modifications  which  may  be  stated  as  follows  : 

enumerated.       jt  ;s  separated  into  several  parts,  which  pursue  different  courses,  or  are  otherwise  differently  modified.     One  of 

these  parts  is  regularly  reflected,  and  pursues,  after  reflexion,  a  course  wholly  exterior  to  the  new  medium,  or  obstacle. 

ru.Hexion  ^  second  and  a   third  portion  are  regularly  refracted,  that   is,  they  enter  the  medium,   and   there   pursue 

their  course    according  to  the    laws   of  refraction.      In  many  media  these   portions  follow  the  same  course 

Regular  re-  precisely,  and   perhaps  are  no  way  distinguishable  from   each  other.      In  such   media   (comprehending  most 

inution.        uncrysiallized  substances  and  liquids)  the  refraction  is  said  to  be  single      In  numerous  others  (for  instance 

single  and    ;„  most  crystallized    media)  they  follow  different  courses,  and  also  retain  different   physical  characters.      In 

&wtion*""     tnese  the  refraction  IS  sa'd  to  °e  double. 

g3  A  fourth  portion  is  scattered  in  all  directions,  one  part  being  intromitted  into  the  medium,  and  distributed 

Scatterin"     over    tne    hemisphere    interior    to    it,  while    the    other   is   in  like    manner  scattered  over   the  exterior  hemi- 
sphere.    These  two  portions  are  those  which  render  visible  the  surfaces  of  bodies  to  eyes  situated  any  how 
with  respect  to  them,  and  are  therefore  of  the  utmost  importance  to  vision. 
g4  Of  those  portions  which  enter  the  medium,  a  part  more  or    less  considerable  is  absorbed,  stifled,  or  lost, 

Absorption,  without  any  further  change  of  direction ;  and  that  not  at  once,  but  progressively,  as  they  penetrate  deeper 
and  deeper  into  its  substance.  In  perfectly  opaque  media,  such  as  the  metals,  this  absorption  is  total,  and 
takes  place  within  a  space  less  than  we  can  appreciate  ;  yet  even  here  we  have  good  reasons  for  believing 
that  it  does  not  take  place  per  saltum.  In  crystallized  bodies,  those  at  least  which  are  coloured,  this  absorp- 
tion takes  place  differently  on  the  two  portions  into  which  the  regularly  refracted  ray  is  divided,  according 
to  laws  to  be  explained  when  we  come  to  treat  of  the  absorption  of  light. 

85.  The  regularly  refracted  portions  of  a  ray  of  white  or    solar    light  are   (except  in    peculiar  circumstances) 
Separation    separated  into  a  multitude  of  rays  of  different    colours,  and  otherwise  differing  in  their   physical  properties, 
into  colours,  each  of  which  rays  pursues  its  course  afterwards,  independently  of  all  the  rest,  according  to  the  laws  of  re- 

on  gular  refraction  or   reflexion.      The  laws  of   this  separation,  or  dispersion,  of  the  coloured    rays,  and  their 
physical  and  sensible  properties,  form  the  subject  of  Chromatics. 

86.  All  those    portions  which  are    either    regularly  reflected,  or    regularly  refracted,  undergo,  more  or  less,  a 
Polarization,  modification    termed   polarization,  in  virtue  of   which  they  present,   on  their   encountering  another  medium, 

different  phenomena  of  reflexion  and  refraction  from  those  presented  by  unpolarized  light.  Generally  speaking, 
polarized  light  obeys  the  same  laws  of  reflexion  and  refraction  as  unpolarized,  as  to  the  directions  which 
the  several  portions,  into  which  it  is  divided  on  encountering  a  new  medium,  take ;  but  differs  from  it  in  the 
relative  intensities  of  those  portions,  which  vary  according  to  the  situation  in  which  the  surface  of  the  medium 
and  certain  imaginary  lines,  or  axes  within  it,  are  presented  to  the  polarized  ray. 

The  rays  of  light  under  certain  circumstances  exercise  a  mutual  influence  on  each  other,  increasing,  dimi- 
Interference  nishing,  or  modifying  each  other's  effects  according  to  peculiar  laws.      This    mutual    influence    is  called  the 
interference  of  the  rays  of  light.     We  shall  proceed  to  treat  of  these  several  modifications  in  order ;  and  first 
of  the  regular  reflexion  of  light. 


352 


LIGHT. 


Light. 


Part  I. 


S8. 


89. 

Laws  of 
reflexion. 

90. 

91. 
92. 


93. 
94. 


95. 

Demon- 
strated by 
experiment, 


96 


97. 


Fig  9. 


98. 

Fie.  10. 


§  3.   Of  the  regular  Reflexion  of  unpolarized  Light  from  Plane  Surfaces. 

When  a  ray  of  light  is  incident  on  a  smooth-polished  surface,  a  portion  of  it  is  regularly  reflected,  and 
pursues  its  course  after  reflexion  in  a  right  line  wholly  exterior  to  the  reflecting  medium.  The  direction 
and  intensity  of  this  portion  are  the  objects  of  inquiry  in  this  section  ;  the  physical  properties  acquired  by 
the  ray  in  the  act  of  reflexion  being  reserved  for  examination  at  a  more  advanced  period.  And  first,  with 
regard  to  the  direction  of  the  reflected  ray.  This  is  determined  by  the  following  laws  : 

Laws  of  Reflexion. 

Law  1.  When  the  reflecting  surface  is  a  plane.  At  the  point  on  which  the  ray  is  incident  raise  a  perpendicular. 
The  reflected  ray  will  lie  in  the  same  plane  with  this  perpendicular,  and  with  the  incident  ray.  It  will  lie 
on  the  opposite  side  of  the  perpendicular,  and  will  make  an  angle  with  it  equal  to  that  made  by  the  in- 
cident ray. 

The  plane  in  which  the  perpendicular  to  any  surface  at  the  point  of  incidence,  and  the  incident  ray,  both 
lie,  is  called  the  plane  of  incidence. 

The  angle  included  between  the  incident  ray  and  the  perpendicular  is  called  the  angle  of  incidence. 

The  plane  in  which  the  reflected  ray  and  perpendicular  both  lie  is  called  the  plane  of  reflexion ;  and  the 
angle  included  between  the  perpendicular  and  reflected  ray  is,  in  like  manner,  termed  the  angle  of  re- 
flexion. 

Adopting  these  definitions,  the  law  of  reflexion  from  a  plane  surface  may  be  announced  by  saying,  that 
the  plane  of  reflexion  is  the  same  with  that  of  incidence,  and  the  angle  of  reflexion  equal  to  that  of  incidence, 
but  situated  on  the  contrary  side  of  the  perpendicular. 

Carol.  The  incident  and  reflected  rays  are  equally  inclined  to  the  surface  at  the  point  of  incidence. 

Law  2.  When  the  surface  is  a  curved  one,  the  course  of  a  ray  reflected  from  any  point  is  the  same  as  if 
it  were  reflected  at  the  same  point  from  a  plane,  a  tangent  to  the  curve  surface  at  that  point  ;  z.  e.  if  a 
perpendicular  be  raised  to  the  curve  surface  at  the  point  of  incidence,  the  reflected  ray  will  lie  in  the  plane 
of  incidence,  and  the  angle  of  reflexion  will  equal  that  of  incidence. 

The  demonstration  of  these  laws  is  a  matter  of  experiment.  If  we  admit  a  small  sunbeam  through  a 
hole  in  the  shutter  of  a  darkened  chamber,  and  receive  it  on  a  polished  surface  of  glass,  or  metal,  we  may 
easily  with  proper  instruments  measure  the  inclinations  of  the  incident  and  reflected  rays  to  the  surface, 
which  will  be  found  equal.  But  this  method  is  rude  and  coarse.  A  much  more  delicate  verification  of  this 
law  is  afforded  by  astronomical  observations.  It  is  the  practice  of  astronomers  to  observe  the  altitudes  of 
the  stars  above  the  horizon  by  direct  vision  ;  and,  at  the  same  instant,  the  apparent  depression  below  the 
horizon  of  their  images  reflected  at  the  surface  of  Mercury,  (which  is  necessarily  exactly  horizontal,)  and  the 
depression  so  observed  is  always  found  precisely  equal  to  the  altitude,  whatever  the  latter  may  be,  whether  great 
or  small.  Now  as  these  observations,  when  made  with  large  instruments,  are  susceptible  of  almost  mathe- 
matical accuracy,  we  may  regard  the  law  of  reflexion,  or  plane  surfaces,  as  the  best  established  in  nature. 

Reflexion  at  a  curved  surface  may  be  considered  as  taking  place  at  that  infinitely  small  portion  of  the 
surface  which  is  common  to  it,  and  to  its  tangent  plane  at  the  point  of  incidence ;  so  that  if  a  perpendicular 
to  the  surface  be  erected  at  the  point  of  incidence,  the  incident  and  reflected  rays  will  make  equal  angles  with  it 
on  opposite  sides. 

Proposition.  To  find  the  direction  of  a  ray  of  light  after  reflexion  at  any  number  of  plane  surfaces,  given  in 
position. 

Construction.  Since  the  direction  of  the  ray  after  reflexion  is  the  same  whether  it  be  reflected  at  the  given 
surfaces,  or  at  surfaces  parallel  to  them,  conceive  surfaces  parallel  to  the  given  ones  to  pass  through  any 
point  C,  (fig.  9,)  and  from  C  draw  the  straight  lines  CP,  C  P',  CP'',  &c.  respectively  perpendicular  to  these 
respective  surfaces,  and  lying  wholly  exterior  to  the  reflecting  media.  Draw  S  C  parallel  to  the  ray  when 
incident  on  the  first  surface,  and  in  the  plane  S  C  P,  and  on  the  opposite  side  of  C  P,  from  the  incident  ray  S  C 
make  the  angle  P  C  s/=  PCS,  then  will  C  /  be  the.  direction  of  the  ray  after  reflexion  at  the  first  surface. 
Prolong  s'C  to  S',  then  S'C  will  represent  the  ray  at  the  moment  of  its  incidence  on  the  second  surface,  whose 
normal  is  C  P'.  Again,  make  the  angle  P'  C  s"  in  the  plane  S'  C  P*,  but  on  the  other  side  of  C  P',  equal  to 
the  angle  S'CF,  then  will  C  s?'  represent  the  ray  at  the  moment  of  its  reflexion  from  the  second  surface,  and, 
producing  «"  C  to  S",  S"C  will  represent  it  at  the  moment  of  its  incidence  on  the  third  surface,  whose  normal 
is  C  P".  Similarly  in  the  plane  S"  C  P" ;  but  on  the  other  side  of  C  P"  make  the  angle  F'  C/"  =  F'  C  S", 
and  C  s'"  will  be  the  direction  of  the  ray  at  the  moment  of  its  quitting  the  third  surface,  and  so  on. 

Analysis.  About  C  as  a  centre  conceive  a  spherical  surface  described,  (fig.  10,)  then  will  the  plane 
P  S  s  intersect  it  in  a  great  circle  P  S  S'  p,  and  the  plane  in  which  C  P,  C  F  lie,  or  the  plane  at  right  angles 
to  the  two  first  reflecting  planes  in  another  great  circle  PP'^7,  and  the  planes  S'Csf'  and  S  Cs"  in  other  great 
circles  S'PVand  Slcs". 

Since  C  P  and  C  P'  are  given  directions,  the  angle  P  C  P',  or  the  arc  PF  (which  is  equal  to  the  inclination 
of  the  two  first  surfaces  to  each  other)  is  given.  Call  this  I.  Again,  since  the  direction  S  C  of  the  incident 
ray  is  given,  the  angle  of  incidence,  or  the  first  surface  P  C  S  (=  a)  and  the  angle  S  P  F,  or  the  inclination  of 
the  plane  of  the  firsl  reflection  to  the  plane  P  P'  perpendicular  to  both  surfaces  (=  fy)  are  given.  Hence  in 


LIGHT.  353 

Li(fhu      the  spherical  triangle  P  F  S'  we  have  P  F  =  I ;  P  S'  =  i  80°  —  a  ;  and  the  angle  P'  P  S'  =  ty  ;  consequently      Part  I. 

—v— '  S'P',  and  therefore  2  S'P'  =  S' s"  and  the  angle  S  S'  F  are  known,   as  also  the  angle  PP'S',   and  therefore  — -  v— • 

its  supplement  P  P's",  which  is  the  angle   made  by  the   second  reflexion  with  the   plane  P  F.     Again,  in  the 

spherical  triangle  S  S' s"  we  have  given  S  S' =  180°  —  2o;  S ' s"  =  2  S' P'   and   the   included    angle  SS's", 

whence  the  third  side  S  *"  may  be  found,  which  is  the  angle  between  the  incident  and  twice  reflected  rays. 

Similarly,  if  a  third  reflexion  be  supposed,  we  have  given  P/  S"=  180°  —  S'P';  P'P"=  P,  and  the  angle 
S"  F  P"  =  S'  F  P"  =  P  P'  P"  -  P  F  S',  whence  we  may  compute  S''  P"  and  proceed  as  before,  and  so  on  to 
any  extent. 

Confining  ourselves  however  to  the  case  of  two  reflexions  we  have,  by  spherical  trigonometry,  putting  P' S'  =         99. 
a'  =   the  angle  of  incidence  on  the  second  reflecting  surface,  P  S'  P'  =  0  ;    P  F  S'  =  0,  and  180°  —  S  s'  =  D, 
the  deviation  of  the  ray  after  the  second  reflexion,  the  following  equations  : 

/  T  •  «      T  General 

—  COS  a!  =  cos  a  .  COS  I  —  sin  a  .  sin  I  .  COS     "  ~-  equations 


sin  I 

sin  0  —   —.  --  r.  sin 


sin  0   = 


nil  i     >«• 

cos  D  =  cos  2  a  .  cos  2  a'  —  sin  2  a. .  sin  2  </ .  cos  0 


of  reflexion 
at  two 


(A)  Planes 


From  these  equations,  any  three  of  the  seven  quantities  a,  a',  I,  0,  0,  y^,  D  being  given,  the  other  four  may       100. 
be  found.     It  will  be  observed,  that  0  is  the  angle  between  the  plane  nf  the  second  reflexion  and  the  principal  Values  of 
section  of  the  two  reflecting  planes,  and  0  the  angle  between  the  planes  of  the  flrst  and  second  reflexion.     If  0  the  symbols 
and  D  only  be  sought,  0  must  be  regarded  as  merely  an  auxiliary  angle  ;    but  this   may  not  be  the  case,  and 
cases  may  'occur  in  which  0  alone  may  be  sought,  or  in  which  it  enters  as  a  given  quantity,  &c.      In  short, 
the  foregoing  equations  contain  in  themselves  all  the  conditions  which  can  arise  in  any  proposed  case  of  two 
reflexions. 

Carol.    If  YT  =  o,  or  if  the  incident  ray  coincide  with  the  principal  seetipn  P  C  P,  i.  e.  if  the  two  reflexions        101. 
both  take  place  in  the  plane  perpendicular  to  the  reflecting  surfaces,  these  fbnnulie  take  a  very  simple  form, 
for  we  then  have 

0  —  0;    0  —   ISO°;    COS  a'  =  —  COS  (a.  +  i) 

that  is  (a  +  a')  =  ISO-  —I  ;  and  consequently  cos  (2  a  +  2  «')  =  cos  (360°  -  2  I)  —  cos  2  I,  or  2  a  +  2  «'  =  2  I. 
Hut  since  6  =  o,  we  have  by  the  last  of  the  equations  (A)  cos  D  =  cos  2  (a  +  «')  ;  consequently  D  =  2  a  +  2  a' 
=  21.     That  is  to  say,  the  deviation  in  this  case,  after  two  reflexions,  is  equal  to  twice  the  inclination  of  the  „ 
reflecting  planes,  whatever  be  the  original  direction  of  the  ray.     This  elegant  property  is  the  foundation  of  the  both  .vile; 
common  sextant  and  of  the   reflecting  circle,  and  is  commonly  regarded   as  having  been  first  applied   to  the  ions   are  in 
measurement  of  angles  by  Hadley,  though  Newton   appears   also   to  have   proposed  it  for  the  same  object,  one  plane. 
See  the  explanation  of  these  instruments. 

In  other  cases,  however,  D,  the  deviation,  is  essentially  a  function  of  the  angles  expressing  the  position  of       ]Q2. 
the  incident  ray,  and  can  only  be  obtained  from  the  equations  above  stated. 

Proposition.  Given  the  angles  of  incidence  on  the  two  planes,  and  the  angle  made  by  the  plane  of  the  first       103. 
reflexion  with  that  of  the  second  ;  required  the  positions  of  the  incident  and  twice  reflected  rays,  the  deviation 
of  the  ray  after  both  reflexions,  and  the  angle  included  between  the  reflecting  surfaces. 

Retaining  the  same  notation,  we  have  given,  a,  a'  0,  required  I,  D,  and  0,  fy. 

1st,  D  is  given  at  once,  by  the  last  of  the  general  equations,  (A.) 

2ndly,  To  find  the  rest,  put  *  =  sin  I  ;    y  =  sin  y-  ;  and  a  —  sin  a'  .  sin  0  ;  put  also  cos  a  =  c  ,•  sin  a  =  s  ; 

cos  a'  =  cf  ;  sin  a'  =  /.     We  have  then  xy  =  a,  or  y  =  --  ;  and  the  first  of  the  equations  (A)  then  gives 

X 


-  tf  =  c  VI  _  x*  —  s  Vxi  —  a* 
which,  cleared  of  radicals  and  reduced,  gives 

0  =  1*  +  j2{2(/2(c2       s")  —  Zc"-  -2s*  a*}  +  (c/2  -  c2)2  +  2  a2  s3  (c'2  +  c2)  +  a*  «« 

and    this   equation,  which,  though  biquadratic,  is  of  a  quadratic  form,  contains   the   general  solution  of  the 
problem. 

Carol.  1.   If   0  =  90°,  or  if  the  planes  of  the  first  and  second  reflexions  be  at  right  angles  to  each   other, 

we  have  simply  sin  I  .  sin  ty  —  sin  a',         and  a  —  siu  </  =  s'. 

In  this  case  our  final  equation  becomes  of  the  two 

reflexions 
O  =  xt  -  2  *2  (1  -  c2  e">)  +   (1  -  C2  c'2)4  are  at  right 

ingles. 
•vhich,  being  a  complete  square,  gives  j?2  =  1  —  c2  c/2. 

No\>  i  =  sin  I,  therefore  r2  =  1  —  cos  Is,  consequently  we  have  the  following  simple  result, 

cos  I  (  =  c  cO  =  COS  a  .  eos  n'  . 

VOL.  iv.  :$  A 


354  L  I  G  H  T. 

Light.      Or  the  cosine  of  the  inclination  of   the   planes  to  each  other  is  equal  to  the  product  of   the  cosines  of   the      I'art  1. 
-_r-_  -,_-  angles  of  incidence  on   each.     And,  vice  versa,  if  this  relation  holds  good,  the  planes  of  the  two  reflexions  will  *• 
necessarily  beat  right  angles  to  each  other;  for,  this  relation  being  supposed,  we  have  of  course  x' =  I  —  c'c", 
and  therefore  1  —  c8  c'2  being  put  for  x*  in  the  general  equation,  the  whole  must  vanish  ;  now  this  substitution 
gives  a  biquadratic  of  a  quadratic  form  for  determining  a,  whicli  must  evidently  be  satisfied  by  taking 

a  =  sin  a',  and  consequently  0  =  90°. 

This  elegant  property  will  be  useful  when  we  come  to  treat  of  the  polarization  of  light. 
105.  Carol.  2.  In  the  same  case  if  0  =  90,  the  deviation  D  is  given  by  the  equation 

cos  D  =  cos  2  a  .  cos  2  a', 

or,  the  cosine  of  the  deviation  is  equal  to  the  product  of  the  cosines  of  the  doubles  of  the  angles  of 
incidence. 

106  Problem.  A  ray  of  light  is  reflected  from  each  of  two  planes  in  such  a  manner  that  all  the  angles  of  inci- 

dence and  reflexion  are  equal.  Given  the  inclination  of  the  planes,  and  the  angles  of  incidence  ;  required, 
first,  the  deviation  ;  secondly,  the  inclination  of  the  planes  of  the  first  and  second  reflexion  to  each  other,  and 
the  angles  made  by  each  of  these  planes  with  the  principal  section  of  the  reflecting  planes. 

Preserving  the  same  notation  we  have  o  =  a',  and  therefore  by  the  third  of  the  equations  (A)  ^  =  0,  so 
that  these  equations  become 

cos  a  (1  +  cos  I)  =  sin  a  .  sin  I  .  cos  ^"i 

sin  a  .  sin  0  =  sin  I .  sin  ty  (a) 

cos  D  =  (cos  2  «)3  —  (sin  2  a)2  .  cos  0      ) 

t         I  \  •  I  I  \ 

107.  The  first  of  these  gives  (putting  for  1  +  cos  I  its  value  2  (  cos  —  j    and  for  sin  I  its  equal  2  .  sin  — -  .  cos  —  I 

cos  ty  =  cotan  «  .  cotan  — ,  (6) 

whence  y/-  is  immediately  known.     Hence  y/-  is  had  by  the  equation 

sin  I 

sin  0  =  —   —  .  sin  Vr.  (c) 

sin  a 

Lastly,  if  we   subtract  each  member  of  the  third  of  the  equations  (a)  from  1,  divide  both  sides  by  2,  and 
reduce,  we  transform  it  into  the  following 

D  0 

sin  —  =   sin  2  a  .  cos    —  .  (rf) 

£  & 

These  equations  afford   ready  and   direct   means  of  computing  T>,  0,  and  D   in  succession,  from  the  known 
values  of  a  and  I ;  the  formula  are  adapted  to  logarithmic  evaluation,  and  are  in  themselves  not  inelegant. 

§  IV.    Of  Reflexion  from  Curved  Surfaces. 

108  '^'ne  reflex'on  °f    a  ray  from  a  curved    surface   is  performed  as  if  it   took   place    at    a   reflecting  plane,  a 

tangent  to  the  point  of  incidence.  The  reflected  ray  will  therefore  lie  in  the  plane  which  contains  the 
incident  ray  and  the  normal  or  perpendicular  at  the  point  of  incidence.  The  general  expressions  for  the 
course  of  the  ray  after  reflexion  at  surfaces  of  double  curvature  being  considerably  complex,  and  not  likely 
to  be  of  great  service  to  us  in  the  sequel,  we  shall  confine  ourselves  to  the  particular  case  of  a  surface  of 
revolution  (comprehending  the  cases  of  a  plane,  and  conoidal  surfaces  of  all  kinds)  where  the  plane  of 
incidence  is  supposed  to  pass  through  the  axis  of  revolution. 

10s).  Proposition.  A  ray  being  incident  on  any  surface  of  revolution  in  a  plane  passing  through  the  axis,  to  find 

General  in-  the  direction  of  the  reflected  ray. 

vestigation         Q  p  (fig    n)  being  a  section  of  the  surface  by  the  plane  of  incidence,  QN  the  axis,  QP  the  incident,  and 
'e  Pr  the  reflected  ray,  which    produced  if  necessary  cuts  the  axis  in  q.      Draw  the  tangent  PT,  the  ordinate 
fleetei)  it      P  M,  and  the  normal  P  N,  which  produce  to  O,  and  put  as  follows, 
any  curve.  , 

F'g-  u-  *  =  QM;7/  =  MP;«  =  — —  ;  e  =  the  angle  M  Q  P, 

a  x 

nr  the  angle  made  by  the  incident  ray  with  the  axis ;  then,  since  the  angle  of  reflexion  is  equal  to  that  of  inci- 
dence, we  have  /  r  P  O  =  O  P  Q,  and  therefore  N  P  q  =  O  P  Q  ;  consequently  Q  PT  =  T  Pa.  Now,  Q  q  = 
QM-M<j=QM-PM.tanMP9 


LIGHT. 


355 


Light.  —  x  —  y  .  tan  {TPM-TPg} 

•—  v  —  '  =  x  _  y  .  tan  {TPM  -TPQ} 

=  x  -  y  .  tan  {TPM-PTM  +  PQM  } 
=  i  -  y  .  tan  {90°  —  2PTM+PQM} 

But  by  the  theory  of  curves  we  have  tan  P  T  M  =  -     -  =  p,  consequently  P  T  M  =  arc  tan  p  =  tan  ~  '  p, 

d  x 

denoting  by  tan~  '  the  inverse  function  of  that  expressed  by  tan;  and  since  P  Q  M  =  0,  this  expression  becomes 
Q  q  =  x  —  y  .  cotan  {  2  .  tan  ~  '  p  —  0  } 

=  x—y.  cotan  {  2  .  tan  ~>      --      -  tan  ~  '     —  (a) 


Part  I. 


/  PM 

(  Because  tan  0  =  —  — 
\  QM 


y  \ 

=  —   1 
x  / 


This  then  is  the  general  expression  for  the  distance  between  the  points  in  which  the  incident  and  reflected  rays 
cut  the  axis. 

Now,  by  Trigonometry,  we  have  (A  and  B  being  any  two  quantities) 

(  2  A  1 

cotan  {  2  tan-1  A—  tan"1  B  }   zscotanjtan-1  —  tan-'B  > 

2  A  -  (1  -  A2)  B 


that  is,  since   cotan  .  tan"'  0  —  —  ,  the  cotangent  and  tangent  being  reciprocals  of  each  other,  simply 

1  -  A*  +  2  A  n 


2A—  (1  -  Aa)  B 

d  y  y 

Applying  this  to  the  present  case,  A  =  —  —  =  p  ;  B  =  —  ,  and  therefore  the  expression  above  found  for  Q  q 


becomes 


_  „ 


2px  —  (\  -  p*)y 

(*  +1>y)    (px-y) 
2pX-(l-p<t)y 


(6) 


These  expressions  contain  the  whole  theory  of  the  foci  arid  aberrations  of  reflecting  surfaces. 
Carol.  1.  To  find  the  angle  made  by  the  reflected  ray  with  the  axis,  which  we  will  call  0'. 
This  is  the  angle  P  q  M,  which  is  the  complement  of  M  P  q.     Now  we  have  found  above 

M  Po  =  90°  —  2  tan-1  p  +  6. 


Hence 


&  —  2  .  tan  -  l  p  -  0 
But  tan  0  =  —  ,  so  that  substituting  we  have 


General  e 
pressions 
for  the 
distance  of 
the  focus 


pojnt  Q? 
110. 

Angle  mad« 
by  the  re- 
fle«ed  ray 
and  the  axis. 


« 


In  all  the  foregoing  formulae  we  have  supposed  the  origin  of  the  x  placed  at  Q  the  radiant  point.  If  we  112 
would  place  it  elsewhere,  as  at  A,  we  have  only  to  write  x  —  a  for  x  throughout.  The  formulae  then  become  Formula? 
on  this  hypothesis,  when  the  ra- 

y  diant  point 

tan  0  =  —  :  (e)  is  not  in  the 

x  —  a  origin  of  the 


,  _    2(x  +  py) 


3  A  :2 


356  L  I  G  H  T. 


If  the  incident  ray  be  parallel  to  the  axis,  we  have  only  to  suppose  the  point  Q  infinitely  distant;  or  placing,       1'urt  1 
as  in  the  last  article,  (he  origin  of  the  x  at  a  point  A  at  a  finite  distance,  to  make  a  (=  AQ)  infinite.     The  ^— — •,— — 


Formute       ab°Ve   exPressions  tnen  give  Q  q  =  GO 

when  the  in- 

cident rays  2  V 

n   i  fan   ft'  — 

are  parallel  .    _      ,, 

10  the  axis.  "  C\ 


1  14.  Proposition.   To  represent  the  incident  and  reflected  rays  by  their  equations. 

The  equation  of  any  straight  line  is  necessarily  of  the  form  Y  =  a  X  +  ft.  Suppose  we  take  A  for  the 
common  origin  of  the  coordinates,  and,  retaining  the  foregoing  notation,  representing  by  x  and  y  the  coor- 
dinates of  the  point  P  in  the  curve,  let  X  and  Y  represent  those  of  any  point  in  the  incident  ray  ;  and,  Q 
being  the  point  in  which  that  ray  cuts  the  axis,  and  AQ  =  a,  it  is  evident,  first,  that  when  X  =  a,  Y  =  o; 
and  secondly,  since  the  ray  passes  through  P,  when  X  =  x,  Y  =  y.  Hence  we  have 

o  =  a  a  +  ft,         and         y  =  a  x  +  ft, 


whence  we  get  a  =  — - —       ft  —  —   —  ;        (1) 

x  —  a  x-a 

therefore,  the  equation  of  the  incident  ray  is 

Y=   -2—  (X-a);       (2) 

X    ~~  (v 

or,  which  is  the  same  in  a  different  form, 

y 

\7~  __      if  _  /"V  *.\    .  /QA 

I      "~~      (/     —       "~  ""       V  ^    ™" "'  ^ )    5  V       ) 

PM  y 


or,  since  tan  0  = 


M  Q  x  —  a 

Y=  (X  -  a).  tan0;          (4) 

or,  again,  Y  —  y  =  (X  —  x)  .  tan  0.         (5) 

Similarly  for  the  reflected  ray,  it  is  obvious  that  if  we  represent  its  equation  by  Y  =  of  X  +  ft',  we  shall  have 

(6) 


~       ,  .--- 

x  —  a  x  —  a! 

and  consequently 

Y  =        --JL-      .  (X  -  a')   =   (X  -  a!)  .  tan  0>  ;       (7) 


.  (X-*)   =  (X-  *).tanfl';       (8) 


, 

dC     ~~     CL 

will  be  the  corresponding  forms  of  the  equation  of  the  reflected  ray,  in  which  a!  and  tan  tf  are  given  in  terms 
of  x,  y,  a,  and  p  =   -—?—  by  the  equations  (g)  and  (A)  or  (i). 

i  iij.  If  the  whole  figure  (fig.  11)  be  turned  about  the  axis  A  M,  and  Q  be  supposed  a  radiant  point,  the  rays  in  the 

Fig.  H.  whole  conical  surface  generated  by  the  revolution  of  QP  will  be  concentred  after  reflexion  in  one  and  ll.e 
same  point  q,  which  will  thus  become  infinitely  more  illuminated  than  by  any  single  ray  from  an  elementary 
molecule  of  the  surface.  The  point  P  will  generate  an  annulus,  having  M  P  for  its  radius  ;  and  q  is  called  the 
focus  of  this  annulus,  and  the  distance  A  q  the  focal  distance  of  the  same  annulus.  This  last  expression  is 
commonly  understood  to  mean  the  distance  of  q  from  the  vertex,  or  point  where  the  curve  meets  the  axis, 

listancc.         ,  J  ., 

Vertex.         Dllt  we  s"all  use  it,  at  present  in  the  more  general  sense. 

116.  Generally  speaking,  then,  the  focus  varies  as  the  point  P  in  the   reflecting  annulus  varies,  unless  in  that 
particular  case    where,  by  the  nature  of  the  curve,  the  function  expressing  a'  is  constant.     Let  us  examine 

117.  this  case. 

Jnvestiga          Proposition.  To  find  the  curve  which  will  have  the  same  focus  for  every  point  in  its  surface  of  revolution,  or 
tion  of  the    on  whjcn  rays  diverging  from  or  converging  to  any  point  Q,  being  incident,  shall  all  after  reflexion  converge 

^hich  re-     to  or  diverge  from  one  point  q. 

fleet  all  the      The  va'ue  of  Q  9  assigned  in  Art.  109,  E  q,  (b)  being  made  constant,  affords  the  equation 

incident 

£tto  °"e  (*+l_y)(P£jr  »>  _    =  constant  =  c. 

P°'m-  2px—  (1  —  p")y 


L  I  G  H  T  357 

Light.          This  equation,  cleared  of  fractions,  and  putting  x  for  x  —  c,  (which  is  merely  shifting  the  origin  of  the  co-      part  / 
^v^»>  ordinates  to  the  distance  c  from  their  former  origin)  becomes  -^  ^^_ 

p  {  x2  —  y-  —  c2  }  =  (1  —  p-)  xy.        (a) 

To  integrate  this  equation,  assume  a  new  variable  z,  such  that  p  y  =  x  z,  and  (multiplying  the  original  equation 
by  y)  we  have  py  (x*  —  y*  —c")  =  xy*  —  x  .  p'-  y", 

that  is  .t  2  (x2  —  y*  —  c2)  =  x y*  —  x3  :-, 

whence  we  find  «*  =    — -   =  Xs  z  —  c'2  .  —    — . 

1+z  1   +  -* 

Differentiating  this  equation  we  get 

dy 


Zy  d  y  (=2pydx  —  zx  zdx  because  p  =    — 


+ 


x"d;  —  c2  d  .   f—  -  —  ) 
V  1  +  2/ 


that  is  x2  rfz  —  c2  d  .  — —   =  o, 

This  equation  may  obviously  be  satisfied  in  two  \vays ;  the  first  is,  by  putting  the  factor 

C"^  C 

which  gives  (restoring  the  value  of  z,  z  =  — —  I  merely  x  +  py  =  c,-  and,  eliminating  p  between  this  and 
the  original  equation  (a)  we  find,  on  reduction, 

r>+  (X-C)*  =  o. 

This  is,  however,  (as  is  clear  from  the  way  in  which  it  has  been  obtained,)  only  a  singular  solution  of  the 
differential  equation,  (see  DIFFERENTIAL  CALCULUS,  singular  solutions;)  and  as  the  value  of  y  which  results 
from  it  is  always  imaginary,  it  affords  no  curve  satisfying  the  conditions  of  the  problem. 

The  other  way  in  which  the  equation  (6)  can  be  satisfied,  is  by  putting  dz  =  o,  or  z  =  constant.     Let  The  curve 

in  all  ruses 

p  y  a  conic 

this  constant  be  represented  by  —  h ;  then,  since  z  =  -    -— ,  we  have  section. 

p  y          y  d  y 
x  x  d  x 

which,  integrated,  gives  y'!  =  h  (a-  —  x"), 

a  being  another  constant.  This  is  the  general  equation  to  the  conic  sections,  and  it  is  obvious,  from  the 
properties  of  these  curves,  that  they  satisfy  the  conditions  ;  because  two  lines  drawn  from  their  foci  to  any 
point  in  their  periphery  make  equal  angles  with  the  tangent  at  that  point,  and,  consequently,  a  ray  proceeding 
from,  or  converging  to,  one  focus,  and  reflected  at  the  curve,  must  necessarily  take  a  direction  to  or  from 
the  other.  But,  the  foregoing  analysis  being  direct,  shows  that  they  possess  this  property  in  common  with 
no  other  curves. 

Thus  in  the  case  of  the  ellipse,  all  rays,  (fig.  12,  )    S  P,  S  F,   &c.  diverging  from  the  focus    S  will  after       |]g. 
reflexion  converge  to  the  other  focus  H,  the  interior  surface  of  the  ellipse  being  polished ;  and  all  rays  Q  P.  Ellipse. 
Q  P*,  &c.  converging  to  S,  will  after  reflexion  diverge  from  H.  *''?•  12- 

In  the  hyperbola,  (fig.  13,)  rays  Q  P,  Q'P,  &c.  converging  to  one  focus  S,  and  incident  on  the  polished  fig.  l.T 
convex  surface  of  the  curve,  will  after  reflexion  converge  to  the  other  focus  H;  and  if  diverging  from  S,  119 
and  reflected  on  the  polished  concave  surface  P  P',  will  after  reflexion  diverge  from  H.  Hyperbola. 

In  the  case  of  the  parabola,  rays  parallel  to  the  axis,  incident  on  the  interior  or  concave  surface,  will  all  be  120. 
reflected  to  the  focus  S,  fig.  14 ;  and  if  reflected  at  the  exterior  or  convex  surfaces,  will  all  after  reflexion  diverge  Parabola. 
from  S.  Fig.  14. 

Rays  converging  to,  or  diverging  from,  the  centre  of  a  sphere  will  all  after  reflexion  diverge  from,  or  \-,\ 
converge  to,  the  same  centre.  Circle. 

Let  us  now  apply  our  general  formula  (b)  (Art.  109)  to  some  particular  cases. 


358  L  I  G  II  T. 

Light.          Pn>pne>/;>>/i.  Let  the  reflecting  surface  be  a  plane,  or  the  curve  PC  a  straight  line.     Required  the  focus  of     Parti. 
**~v^s  reflected  rays.  s_—  v.™. 

Focus  of  a        Here  we  have  x  =  constant  =  a  p=  -      -  =    cc  ,  and  the  general  formula  becomes  simply 
plane  sur- 
face. 2  x  y 

Q  q  =  a  =  —     - —  =  2  x  =  2  a. 

y 

So  that  the  focus  of  reflected  rays  is  a  point  on  the  opposite  side  of  the  reflecting  plane  equally  distant  from 
it  with  the  radiant  point;  and  as  this  is  independent  of  y,  or  of  the  situation  of  the  point  P,  we  see  that  all 
the  rays  after  reflexion  diverge  from  this  point,  see  fig.  la. 

123.  Proposition.  To  find  the  focus  of  any  annulus  of  a  spherical  reflector. 

Focus  of  a        Let  r  be  the  radius  of  the  sphere,  and,  if  we  fix  the  origin  of   the  coordinates  at  the   radiant    point,  t'.ia 
spherical      equation  of  the  generating  circle  will  be 

annului. 

ra=  O-  o)°-  +  y* 
This,  differentiated,  gives  (x  —  a)  d  x  +  ;/  dy  =  o, 

d  y  x  —  a  2  «4  —  r°- 

consequently  p  =  — ; — •    =  —  • ;    1  —  P2  =  . 

dx  y  3/2 

Hence,  substituting  in  the  general  expression  (6),  we  find  for  the  focal  distance  the  following  value, 


r*  4  2  a  (x  -  a) 

which  expresses  in  all  cases  the  distance  of  the  focus  of  reflected  rays  from  the  radiant  point. 

For  optical  purposes,  however,  it  is  more  convenient   to  know  its  distance  from  the  centre,  or  from  the 
surface. 

The  distance  from  the  centre  (E  q,  fig.  16,)  is 

2  a  (ax-  a°-  +  r«) 


2  a  (r  —  a)  +  r8 

in  which  positive  values  of  E  q  lie  to  the  right  of  E,  or  the  same  way  with  those  of  d  or  of  Q  q. 
Focus  for          Carol.  1     If  we  would  find  the  focus  of  the  infinitely  small  annulus  immediately  adjoining  to  the  vertex  C, 
central  rays  or  C'  of  the  reflecting  spherical   surface,  or,   as  it  is  termed  in  Optics,  the  focus  of  central  rays,  we  must  put 
in  a  sphe-    ;n  the  case  of  the  vertex  C  (when  the  reflexion  takes  place  on  a  concave  surface)  x  =  a  +  r,  and  in  the  other 

ncal  re-       case  viz.  that  where  the  rays  are  reflected  on  the  convex  surface  C',  x  =  a  —  r.     The  former  gives 
Hector. 


the  latter  gives  the  same  results,  writing  only  —  r  for  r. 

124.  If  we  bisect  the  radii  C  E  and  C'  E  in  F  and  F',  and   suppose  q  and  q'  to  be  the  foci  of  central  rays  reflected 

(JL)1 

at  C  and  at  C',  we  shall  have  F  q  =  i  r  —  -  -    =  -  -  ,       (d) 

2a  +  r  r 

a   +     ¥ 

which  gives  the  following  useful  analogy, 

QF  :  FE  :  :  EF  :  Fq.  (e) 

Similarly  we  have  QF*  :  F'E  :  :  E  F'  :  F'  q  ;  so  that  the  same  analogy  applies  to  both  cases,  and  may  be 
regarded  as  the  fundamental  proposition  in  the  theory  of  the  foci  for  central  rays.  For  it  is  obvious,  that  if  P  C 
were  any  other  eurve  than  a  circle,  the  same  must  hold  good,  taking  only  E  the  centre  of  curvature  at  the 
vertex. 

125.  Carol.  2.  If  a  be  infinite,  or  the  incident  rays  be  parallel,  we  have  F  q  =  o,  which   shows  that  the  fonts  nf 
Principal      central  parallel  rays  bisects  the  radius.     This  focus,  for  distinction's  sake,  is  called  the  principal  focus  of  the 
focus.           reflector. 

126.  Definition.     Q  and  q  are  termed  conjugate  foci.     It  is  evident  that  if  q  be  made  the  radiant  point,  Q  will 
Conjugate     be  its  focus  ;   for  the  rays  will  pursue  the  same  course  backwards. 

foci.  Carol.  3.  Regarding  only  central  rays  :    the  conjugate  foci   move  in  opposite  directions,  and  coincide  at  the 

1  27-       centre  and  surface  of  the  reflector. 

For  let  a  vary  from  x  to  —  CD  ,  then  F  q  will  vary  as  follows  :  first,  while  a  varies  from  x    to  -      —  F,  q  is 


LIGHT.  359 

Lijht.      positive,  and  increases  from  o  to   CD  ;  that  is,  as  Q  moves  up  to  F,  q  moves   through  C  to  infinity.     As  the      IJan  1 

motion  of  q  continues,  Fq  then  becomes  negative;  because  a  is  then  negative  and  greater  than  -^— ,  and  a   in-  Conjugate 

foci  move 

creasing  Fq  diminishes  ;    therefore  q  moves  from  the  right  towards  F,  that  is  in  the  opposite  direction  to  Q's  '"  °PPosit(: 
motion  ;   and  when  Q  is  at  an  infinite  distance  to  the  right,  q  is  again  at  F. 

When  Q  comes  to  E,  a  =  o  .  F  q  =  —,  or  9  is  at  E  also. 

r 
When  Q  comes  to  C,  a  =  —  r,  F  q  =  —  -— - ,  or  q  is  at  C  also. 

It  appears  by  the  value  of   Eq,  Equation  (b),  that  a  spherical    reflector  ACB,  fig.  17,  whose  chord  (or      128 
aperture,  as  it  is  termed  in  Optics)  is  A  B,  causes  the  ray  reflected  at  its  exterior  annulus  A  to  converge  to,  or  Longitudi- 
diverge  from,  a  point  q,  different  from  the  focus  of  central  rays.     Let  /  be  this  latter  focus,  then  we  shall  have  nal  aberra- 
tion, for  any 
_,,,  a  r  (a  +  r)  r  a  r"  a  r  aperture. 

=   2TTT'   C/=     -2TT^;  fq  =    8«(*-«)  +  r«      '    2^+V  Fig'  I7' 

This  quantity  fq  is  called  the  longitudinal  aberration  of  the  spherical  reflector.     If  the  rays  fall  on  the  convex 
portion,  we  need  only  write  —  r  for  r. 

Proposition.  To  express  approximately  the  longitudinal  aberration  of  :i  spherical  reflector  whose  aperture  is      129. 
inconsiderable  with  respect  to  its  focal  length.  l.ongitudi- 

, y 2  nal  aberra- 

y  being  the  semi-aperture,  and  a  —  a  being  equal  to    v  r*  —  y  2  ---  i          -2 —  ,  (neglecting  y\  and  higher  tion  for 

*  r  small 

powers  of  y,)  we  have  apertures. 

a  r2                              a  r                   a2  y- 
J  q  =  aberration   =   • —   ; — — —  ;         (f) 


J*    ({f,     -J-    f\ 

If  we  put  Cf—f,  we  have  /=  — —      — ,  and,  consequently,  we  may  eliminate  a,  the  distance  of  the      130 

2  a  4-  r  . 

Another 

radiant  point,  and  express  the  aberration  in  terms  of  the  aperture,  radius  of  curvature,  and  distance  of  the  focus  exPresslon 

r  /r f\ 

of  central  rays  from  C,  the  vertex  of  the  minor ;  for  this  gives  a  =    —£j — — i  which,  substituted  for  a,  in 

J      "~     ' 

the  expression  (f)  gives 

(r  —  /) « .  y  2  E  /"* .  (semi-aperture)  2 

aberration  =  — ^—- — 2—   =  — '- — .        (g) 

r3  (rad.)3 

To  express  the  lateral  aberration,  or  the  quantity  by  which  the  reflected  ray  A  q  g  deviates  from  the  axis,  at      131. 
the  focus  of  centra]  rays,  or  the  value  of  fg,  (fig.  17,)  we  have  Lateral ' 

.  -»,  aberration 

fg  =  fl-  -       — ;  but  AM  =  y.and  M$  =  E  M—  E  y  =  *  —  a  —  - —   ar       — 

J*i  q  a  a  (x  —  a)  +  r* 

2  a  (x  —  a)  s  +  r2  (x  —  2  a) 


2a  (x  —  a)  +  r2 
r  a  —x  +  r 


;  so  that 

(h) 


'   2a  +  r  r*(x  +  2  a,  +  2  a  (x  -  a)*  ~ 

When  the  aperture  is  very  small,  this  becomes  simply  132. 

ai  ,,3  Lateral 

f  s  —  9 (f)  aberration 

J  h          r4  .  (r  +  a)  (r  +  2  a)  for  small 

apertures. 

When  a  is  infinite,  or  the  incident  rays  are  parallel,  we  have  the  following,  133. 

s  Aberrations 

/,  =  longitudinal  aberration   =   ^-      )  j^f 

V,          (f\  small 

,                                                                    V3         I  apertures. 
/  g  =  lateral  aberration              = 

If  the  rays  fall  on  the  convex  side  of  the  sphere  we  must  make  r  negative,  which  only  changes  the  signs  of 
the  aberrations. 


360  LIGHT. 

U^L  Part  I. 

§  V.   Of  Caustics  by  Reflexion,  or  Catacaustic.-. 

131  If  rays  of  light  be  incident  on  a  medium  of  any  other  form  than  that  of  a  conic  section,  having  the  radiant 

point  in  the  focus,  they  will  after  reflexion  no  longer  converge  to  or  diverge  from  any  one  point,  but  will  be 

dispersed  according  to  a  law  depending  on  the  nature  of  the  reflecting  curve  ;   the  inclination  of  each  reflected 

ray  to  the  axis  varying  according  to  the  point  of  the  curve  from  which  it  is  reflected,  and  not  being  the  same 

for  any  two  consecutive  rays.     Of  course  each  lay  will  intersect  that  immediately  consecutive  to  it  in  some  point 

or  other,  and  the  locus  of  these  points  of  continual  intersection  will  trace  out  a  curve  to  which  the  reflected  rays 

dustics  by  will   all  necessarily  be  tangents,  and  which   is  called  a  caustic.     If  these  rays  fall  on  another  reflecting  curve, 

reflexion      tney  wju  j)e  a;rajn  dispersed,  and  another  caustic  will  originate  in  the  continual  intersections  of  the  consecutive 

rays  of  the  former,  and  so  on  to  infinity. 

135.  Let  Q  P,  Q'  P',  (fig.  18,)  be  any  two  contiguous  rays  incident  on  consecutive  points  P,  P'  of  a  reflecting  curve 
Fig.  18.       pp',  and  after  reflexion  let  them  pursue  the  courses  P  R,  P' R' ;  and  since  they  are  not  necessarily  parallel, 

let  Y  be  their  point  of  intersection,  then  will  Y  be  the  point  in  the  caustic  Y  Y'  Y''  corresponding  to  the 
point  P  in  the  reflecting  curve ;  and  if  we  determine  the  points  Y'  Y",  &c.  from  the  consecutive  points  P'  P/(,  &c. 
in  the  same  manner,  the  locus  of  these,  or  the  curve  YY' Yx/will  be  the  whole  caustic. 

136.  Since  the  reflected  ray  passes  through  P,  whose  coordinates  are  xy,  its  equation,  as  we  have  already  seen 
Coordinates ,( Art.  114),  is  necessarily  of  the  form 

caustic  in-  ^         "U  =  *    (^  —  X) 

on  any  sup-  If vve  regard  x,  y,  P  as  variable,  this  will  represent  any  one  of  the  reflected  rays  P  R,  and  the  consecutive  ray 
portion  of  P'  R'  will  be  represented  by 

Y  -  (y  + d  y)  =  (p  +  d  p)  (x  -  (*  + d ')  > 


Now  since  the  point  Y  in  which  these  two  rays  intersect  is  common  to  both,  the  coordinates  X  and  Y  at  this 
point  are  the  same  for  both  ;  and  therefore  at  this  point  both  these  equations  coexist,  and  thereby  determine  the 
values  of  X  and  Y,  or  the  situation  of  the  point  Y.  Now  the  latter  of  these  equations  is  nothing  more  than 
the  former  plus  its  differential,  on  the  supposition  of  X  and  Y  remaining  constant.  Therefore,  we  have  to  find 
X  and  Y  from  the  two  equations, 

—  dy  =  (X-x)dP-Fdz, 
•hich  gives  at  once 


O  ij]    (  £    _    fj\    _    /"I     _    /r\  2\   y 

In  these  equations  we  have  only  to  substitute  for  P  its  value  =  tan  Of,  or  —  —  --  '-  -  -  -  *     y     ;   and 

(1  -p9)  (x-a)  +2py 

after  executing  all  the  differentiations  indicated,  or  implied,  to  eliminate  x  and  y  by  the  equations  of  the  curve 
and  the  other  conditions  to  which  the  quantity  a  may  be  subjected,  an  equation  between  X  and  Y  will 
result  which  will  be  the  equation  of  the  caustic. 

137.  Proposition.    To  determine  the  caustic  when   rays    diverge    from   one  fixed  point  in  the  axis  of  a  given 

Caustic        reflecting  curve. 

when  rays        jn  this  case  a  is  invariable,  and  the  differentiation  of  P  must  be  performed  on  this  hypothesis.     It  will, 
>m  therefore,  simplify  the  question  if  we  put  a  =  o  ;  or  suppose  the  origin  of  the  coordinates  in  the  radiant  point, 
point.  in  which  case 

p_    2px-  (l-p')y 

,  (1 


2py  +  (1  -  p*)x 
+  p*)(y-px)  + 


dx 

(I) 

Where  q  =  -^- 

d  x 

p  (!  +  !»•)  (!»*-») 

•2py  +  (1   -pa)  x 


Mght.      wnich  substituted,  we  find 


LIGHT. 

x  —  y)2  —  qx(x*  +  y*) 


361 


Pnrt    I. 


«)  (px-y)-  2 


Y  =  2  . 


-  (1  +  p 

Carol.  \.  If  the  incident  rays  be  parallel,  or  the  radiant  point  at  an  infinite  distance,  we  may  fix  the  origin       138. 
of  the  coordinates  where  we  please  ;  and  since  in  this  case  the  equation  of  any  reflected  ray  is,  by  1 13  equation  Caustic  for 
(;')  and  114  equation  (8),  parallel 

2  D  ra)'s- 

Y  -  y  =  (X  -  X)  .  r_L-.  (TO,  2) 


1  -p'' 


we  have 


P  = 


! ;  P  —  p  =  _— - 

l-p«  "  '  1  - 


dp             d ^  y 
putting  q  for  —    or — . 


These  substitutions  made,  we  get  the  following  values  for  the  coordinates  of  the  caustic, 

X  =  *  +  -£-(l-s)-Y  =       -      r'°~ 
2q  '  q     ' 


00 


Carol.  2.  In  the  general  case,  if  we  put  /  =  the  line  P  y,  or  the  distance  between  the  point  in  the  curve       ioq 
and  the  corresponding  point  in  the  caustic  we  have  n-  ** 

-  . between 

/==    "*    (X  —   x)2   +   (Y  —  y)*  correspond- 

Which,  if  we  write  for  X  —  x  and  Y  —  y,  their  values  above  found  become  in  curve 

and  caustic 


or,  writing  for  P  its  value,  and  executing  the  operations, 

,_          -^(yj-px)  (1  +  ;,') 


(y  -  p  J")  (1 
Carol.  3.  In  the  case  of  parallel  rays,  when 


+  2 


+ 


140. 


we  have 


/  ) 


t°he 


9  -v^jr2  +j/"J 


so  that 


,  (,«  +  y.)  =  _2_(P 


and  substituting  this  for  9  (r2  +  y'),  in  the  general  expression  for  /,  we  eliminate  9,  and  get 

,_          c/j?"  +  y*  rc 


4r-c 


putting 
Hence  we  have 


r  — 


(r) 


STnd  fhJn^twoL^T.1  Pfr0f?erty-     (Smith>s  Optic*,  ed.   173ft,  p.  160.)  ,„. 

"'  °f  a,"  e'ementary  pencil  of  ravs  reflected  at  any  curve  surface  at  P,  fig.  19    tig.  19 
'- ;    (if  the  curve  be  a  circle,  this  will   be  the  curve  itself.)     Let  the  chords 

3  a 


LIGHT. 


Light      PV,  PWin  the  direction  of  the  incident  and  reflected  rays  be  divided  in  F, /,  so  that  PF  and  P/ shall  each 
v— v— •  be  one  quarter  of  the  whole  chords,  and  the  relation  between  Q  and  9  will  be  expressed  by  the  proportion 
General 

QF  •  FP  ::  P/:/9.          (*) 

P-P 


relation 
between 
conjugate 
points  or 
foci  of  re- 
flected rays 
incident  on 
any  curve. 
143. 


Carol.  5.  Putting 


d  x.  =  M,  we  have 


+  M 


dX 
dx 


=  1  + 


dx 


dM 


Hence  it  follows  that 


P  = 


dx 

dY 
~rf~X~ 


d  M 
dx 


P  therefore  is  to  the  caustic,  for  the  coordinates  X,  Y,  what  <p  is  to  the  reflecting  curve  foi   the   corresponding 

point  whose  coordinates  are  x,  y. 

144.  Carol.  6.  If  we  put  S  for  the  length  of  the  caustic 

Length  of 
the  caustic 
investigated. 


or 


because 


=  the  arc  A  H  K  Y,  we  have  d  S  =  •/  d  X * 
d  S  =  d  X .  v'l  +  p*  =  (dx  +  dM)  «/T 

-7-TT,    .  PdP 


dx. 


df=  d  .  M  .  Vl  +  P1  +M. 


+Pa 


PrfP 


— -  ;  but  M  d  P  —  (P  -  p)  d  t 


so  that  we  have 


that  is,  substituting  for  P  its  value 


=  df+  dx. 


1  +  Pp 


V  I  +  P" 
2px-  (1  -  p1)  y 


•2py 


dS=df+  dx  .     -  _J. 


=df+d. 

J 


and  integrating 


S  =  constant  +  /  +   ^x1  +  y*. 


Caustics       Hence  it  follows,  that  the  caustic  is  always  a  rectifiable  curve,  and  its 
always  rec- 

"fiab'e  Length  A  Ky  =   QP  +  Py  +  constant  | 


But 


Arc        A  K  F  =  Q  C  +  C  F  +  constant  j 


consequently,  subtract!  117 


Arc        Fy       =  (QC  +  CF)  -  (QP  +  P  Y). 


145. 

F.g.  20. 


146. 

General 
relation  be- 
tween two 
conjugate 
caustics 
ind  their 
intermediate 
redacting 
curve  inves- 
igated. 


Hence  it  appears,  that  the  caustic  is  necessarily  a  rectifiable  curve  when  the  reflecting  curve  is  not  itself 
transcendental. 

If  the  rays  PR,  P' R',  P"  R",  &c.  after  reflexion  at  the  curve  PFP"fall  on  another  reflector  RR'R"  and 
are  reflected  in  the  directions  R  S,  R'S',  R"  S",  &c.  (fig.  20)  their  continual  intersections  will  form  another 
caustic  Z  Z' Z",  and  so  on  ad  iiifinitum,  which  may  be  determined  by  a  similar  analysis.  In  like  manner, 
whatever  be  the  law  according  to  which  the  rays  Q  P,  Q'  P',  &c.  are  dispersed,  we  may  conceive  each  to  be  a 
tangent  to  a  curve  which  may  be  regarded  as  the  caustic  of  another  reflecting  curve,  and  so  on.  Let  V  V'V" 
be  this  curve.  Since  P  V  Q  is  a  tangent  to  it,  if  this  curve  and  the  curve  P  P'  P"  be  given,  the  point  Q  in 
the  axis  from  which  the  incident  ray  Q  P  may  be  regarded  as  radiating,  is  determined  in  terms  of  the  co- 
ordinates of  P,  and  therefore  the  quantity  a  may  be  eliminated  altogether.  The  manner  of  doing  this  is 
shown  in  the  following 

Proposition.  To  determine  the  relations  between  any  two  consecutive,  or,  as  they  may  be  termed,  conjugate 
caustics  VV  V,  YY'Y'',  and  the  intermediate  reflecting  curve  PP'P". 

Let  V  and  Y  be,  as  before,  any  two  conjugate  points  in  the  caustics,  P  the  reflecting  point ;  then  if  we  put 

f  and  if  for  the  coordinates  of  V 
x  and  y  for  those  of  P 

X  and  Y  of  Y 


LIGHT. 

Light      Since  the  line  P  V  Q  is  a  tangent  to  the  first  curve  at  V,  we  must  evidently  have 


and  this,  combined  with  the  equation  between  i;  and  f,  which  represents  the  curve  V  V  V"  suffices  to  determine 
i]  and  f  in  terms  of  x,  y,  or  vice  versd,  x  and  y  in  terms  of  f  and  17. 
Again,  we  have  also  by  Art.  114,  equation  (2) 

y-'>  =  -rra  (*-f) 

and  consequently 


y-i  y  -i 

Thus  a  is  given  in  terms  either  of  x,  y,  or  of  rj,  f,  whichever  we  may  prefer.     It  only  remains  to  substitute  this 
in  the  value  of  P. 


which  thus  becomes 

p_ 


(1  -  p«)  (x  -  0  +  2  p  (y  -  »,)    1 

and  this,  being  free  of  a,  may  be  substituted  in  the  equations  (&)  Art.  136,  when  X  and  Y  will  be  at  once  ob- 
tained in  terms  of  x,  y,  f,  7,  the  coordinates  of  the  reflecting  curve  and  the  preceding  caustic. 

We  shall  now  proceed  to  illustrate  the  theory  above  delivered  by  an  example  or  two. 

Required  the  caustic  when  the  reflecting  curve  is  a  cycloid,  and  the  incident  rays  are  parallel  to  each  other       147. 
and  to  the  axis  of  the  cycloid.  Caustic  of 

d  y  *S  a  cycloid- 

The  equation  of  the  cycloid  is  -  —  =  p  = 

d  x 


_  x 


taking  unity  for  the  radius  of  the  generating  circle. 
From  this  we  get 

— (2  -  x) 


and  therefore  —  =  2  x  —  x' ; 

q 

consequently,  by  the  equations  (k)  of  Art.  136,  we  shall  have 


whence 

dY 


dx  Vz^ 

Now  we  have  also 


x  -  x'  =  2 


dx 

But  since  X  =  2  x  —  x',  we  have  1—  x  =  ^  1  —  X,         and  therefore 

dX 


dx 


=  2  A/1  -X 


/7V  /      Y 

So  that  we  have,  finally,  =  \  /  — - — — 

a  X  »     i  —  x 

which  shows  that  the  caustic  is  itself  a  cycloid  of  half  the  linear  dimensions  of  the  reflecting  curve.  Is  itself 

To  take  one  other  example,  let  us  suppose  the  reflecting  curve  a  circle,  and  the    radiant   point  infinitely  cyc|oi()- 
distant.     Here  we  have  (placing  the  origin  of  the  coordinates  in  the  centre) 

3  B  2 


f  _     f 

~ 

Caustic  of  »  consequently,  by  the  equations  (A)  of  Art.  136 
-«=le-  P(1 

~ 


LIGHT. 

r*  Pan  I. 


Then  since  (supposing,  for  brevity,  r  =  1,  which  will  not  affect  the  result) 

+  4  xe 


4  Ye=  4-  12  xs  +  l-2xi-4x, 

Adding-,  4  (Xs  +  Y2)=  4  -Si8;  a-'  =  -i  (1  -  X8  -  Y") 

3 

So  that  we  get,  finally,  substituting  this  value  of  x*  in  that  of  Y,  and  reducing, 

(4X»+  4Y2-  I)3  =27  Y";  (») 

which  is  the  equation  of  the  caustic. 

This  equation  belongs  to  an  epicycloid  generated  by  the  revolution  of  a  circle  whose  radius  is  -j  that  of  the 
reflecting  circle  on  another  concentric  with  the  latter,  and  whose  radius  is  £  that  of  the  reflecting  circle. 
.  ,j  Fig.  21  represents  the  caustic  in  this  case;  QP  being  the  incident  ray,  and  P  Y  the  reflected.  It  has  a  cusp 
at  F,  which  is  the  principal  focus  of  rays  reflected  at  the  concave  surface  BCD,  and  another  at  F',  which  is 
that  of  the  rays  reflected  from  the  convex  surface  BAD.  In  the  latter  case,  it  is  not  the  rays  themselves, 
but  their  prolongations  backwards  which  touch  the  caustic. 

149.  Corol.    When    y  is  very  small,  or  immediately  adjacent  to  the  cusp  F,  the  form  of  the  caustic  approaches 

indefinitely  to  that  of  a  semicubical  parabola.     For,  generally, 


X  =  1  -/l  +  3Y1-4  Y", 
and  when  Y  is  very  small,  neglecting  Y8  in  comparison  with  Y  3 


It  is,  as  we  have  seen,  only  in  certain  very  particular  cases,  when  rays  proceeding  from  one  point  and  reflected 
at  a  curve  proceed  after  reflexion  all  to  or  from  one  point.  In  general  they  are  distributed  in  the  manner 
described  in  Art.  145,  146,  being  all  tangents  to  the  caustic.  The  density  of  the  rays  therefore  in  any  point  of  the 
caustic  is  infinitely  greater  than  in  the  space  surrounding  it,  and  in  the  space  between  the  caustic  and  the  re- 
flecting curve  (PCFY,  fig.  18)  is  greater  than  in  the  space  without  the  caustic  Q  YF.  This  is  obvious,  for 
in  the  latter  space  only  the  incident  rays  occur,  while  in  the  former  are  included  all  the  reflected  rays  as  well 
as  the  incident  ones. 

,E|  This   may  be  easily  shown  experimentally,  in  a  very  satisfactory  manner  pointed  out  by  Dr.  Brewster,  by 

"  22  bending  a  narrow  strip  of  polished  steel  into  any  concave  form,  as  in  fig.  22,  and  placing  it  upright  on  a  sheet 
of  white  paper.  If  in  this  state  it  be  exposed  to  the  rays  of  the  sun,  holding  the  plane  of  the  paper  so  as  to 
pass  nearly  but  not  quite  through  the  sun,  the  caustic  will  be  seen  traced  on  the  paper,  and  marked  by  a  very 
bright  well-defined  line  ;  the  part  within  being  brighter  than  that  without,  and  the  light  graduating  away  from 
the  caustic  inwards  by  rapid  gradations.  If  the  form  of  the  spring  be  varied,  all  the  varieties  of  catacaustics, 
with  their  singular  points,  cusps,  contrary  flexures,  &c.  will  be  seen  beautifully  developed.  The  experiment  is 
at  once  amusing  and  instructive. 

The  bright  line  seen  on  the  surface  of  a  drinking-glass  full  of  milk,  or,  better  still,  of  ink,  standing  in  sunshine, 
is  a  familiar  instance  of  the  caustic  of  a  circle  just  investigated. 

152.  If  the  figure  18  be  turned  round  its  axis,  the  reflecting  curve  will  generate  a  surface  of  revolution,  which,  if 

supposed  polished  within  or  without,  as  the  case  may  be,  will  become  a  mirror.  The  caustic  will  also  generate 
a  conoidal  surface,  to  which  all  the  rays  reflected  by  the  mirror  will  be  tangents.  No  mirror,  therefore,  which 
is  not  formed  by  the  revolution  of  a  conic  section  having  the  radiant  point  in  its  focus,  can  converge  all  the 
reflected  rays  to  one  point  or  focus.  There  will,  however,  always  be  one  point  which  receives  the  reflected  rays 
in  a  more  dense  state  than  any  other.  This  point  is  the  cusp  F,  as  we  shall  presently  see.  The  deviation  of 
any  reflected  ray  from  this  point  is  what  is  termed  its  aberration. 

i  c,o  The  concentration  and  dispersion  of  rays  by  reflecting  and  refracting  surfaces  forming  the  great  business  of 

practical  optics,  it  will  be  necessary  to  enter  at  large  into  this  subject  ;  and,  first,  it  will  be  proper  to  inquire 
how  far  any  given  reflector  will  enable  us  to  concentrate  the  rays  which  fall  on  it.  To  this  end  let  the  following 
problem  be  proposed. 

154  Proposition.  A  reflector  of  any  figure,  of  a  given  diameter  or  aperture  AB,  being  proposed,  to  find  the  circle 

of  least  aberration,  or  the  place  where  a  screen  must  be  placed  to  receive  all  the  rays  reflected  from  the  surface, 
within  the  least  possible  circular  space  (since  they  cannot  be  all  collected  in  one  point)  and  the  diameter  of  this 
circle. 


L  I  G  H  T.  365 

Light.          AC  B  (fig.  23)  being  the  mirror,  Q  the  radiant  point,  GKfkg  the  caustic,  /the  cusp  or  focus  for  central      Part  1. 
— v— — '  rays,  q  the  focus  of  the  extreme  rays  A  q,  B  q,  produce   these  lines  till  they  cut  the  caustic  in  Yy.     It  is  c.ear,  v~^^--~~* 
then,  since  all  the  rays  reflected  from  the  portion  A  C  B  of  the  reflector  are  tangents  to  points  of  the  caustic  F'R-  23- 
between  K,/and  k,f,  that  they  must  all  pass  through  the  line  Yy.    Retaining  the  notation  of  the  foregoing  pro- 

o  o  o  o 

positions,  (i.  e.  supposing  Q  x  =*  X  ;  X  y  =  Y.)     Let  us  put  QL  =  X,  L  K  =  Y,  QD  —  x;  D  A  =  y  ;  and  let 

P,  p  represent  the  values  of  P  and  p  corresponding  to  the  points  K  and  A  of  the  caustic  and  reflecting  curves. 
The  equation  of  the  line  A  K  9  y  will  then  be 

Y-y=F(X-.J);  (*) 

Y  and  X  being  the  coordinates  of  any  point  in  it.  But  at  the  point  y,  where  it  cuts  the  other  branch  of  the 
caustic,  these  coordinates  are  common  to  the  straight  line,  and  to  the  caustic.  At  this  point,  therefore,  the  above 
equation,  and  those  expressing  the  nature  of  the  caustic,  must  subsist  together.  Now  these  are  the  equations 
(k)  Art.  136,  combined  with  the  original  equation  of  the  reflecting  curve.  Eliminating,  then,  x  and  y,  by  the 
aid  of  two  of  them,  and  determining  the  values  of  X,  Y  from  the  rest,  the  problem  is  resolved. 

Now  the  same  equation  which  gives  the  value  of  y,  or  xy,  must  also  give  that  of  LK,  because  K  is  a  point       155. 
in  both  caustic  and  the  line  A  K  y,  as  well  as  y.     But,  moreover,  since  A  K  y  is  a  tangent,  the  point  K  is  a 
double  point ;  therefore  the  final  equation  in  Y  must  necessarily  have  two  equal  roots,  besides  the  value  of  Y 
sought ;  and  these  being  known,  the  other  may  be  found  from  a  depressed  equation. 

The  method  here  followed  is,  apparently,  different  from  that  usually  employed,  which  consists  in  making  the 
value  of  y  as  determined  by  the  intersection  of  the  extreme  reflected  ray  AKy,  and  any  other  reflected  ray  (from  P) 
a  maximum.  But  the  difference  is  only  apparent,  for  in  the  latter  method  we  have  to  make  Y  as  determined 
by  the  two  equations  (holding  good  jointly) 

Y  -  y  =  P  (X  -  x),  and  Y  -  y  =  P  (X  -  x) 

a  maximum,  or  dY  =  o.  Now  in  this  case  the  former  equation  gives  dX  =  o  also;  and  therefore,  differen- 
tiating the  latter,  we  have  —  d  y  =  (X  —  x)dP  —  P  dx, 

P  —  r> 
whence  X  —  x  = —  d  x 

P  —  v 

and  therefore  Y  —  y  —  P  . d  x. 

Now  these  are  nothing  more  than  the  equations  of  Art.  136,  expressing  the  general  properties  of  the  caustic; 
so  that  this  consideration  of  the  maximum  only  leads  by  a  more  circuitous  route  to  the  same  equations  as 
the  method  above  stated,  and  is  in  fact  nothing  more  than  a  different  mode  of  expressing  the  caustic. 

Let  us  apply  this  reasoning  to  the  case  when  the  reflector  is  spherical.  Resuming  the  equations  and  156. 
notation  of  Art.  148,  and  putting  a  for  the  extreme  value  of  y,  or  the  semi-aperture  of  the  mirror,  and  6  for  tircle  of 
the  corresponding  value  of  x,  that  of  P  will  be 

ration   in  a 

•2  a  b                   2ab  sPHherical 
—    . reflector. 


i-  - )  - 


Hence  the  equation  (m,  2)  Art.  138,  of  the  extreme  reflected  ray  becomes 


whence  we  get  2  X  =  —  ( \    +    — — — - —  . 

6     \  a 

Assume  z,  so  that  Y  =  as  z3,  z  being  another  unknown  quantity,  then  we  have 

4X4  =  i-        i    U+d  -2«')a'2'}!. 

i   ~^~  \L 

Substituting  this  for  4  Xs,  and  for  Y*  its  value  a"  z"  in  the  equation  of  the  caustic  (»)  Art.  148,  extracting  the 
cube  root,  and  reducing,  we  get  the  following  equation  for  finding  z, 

a*z'  +  (2-  4 a")  z3  +  (3  «8  -  3)  z    +  1  =  o. 

Now  this,  according  to  the  remark  in  Art.  155,  must  have  two  equal  roots,  viz.  when  x  =  b,  or  Y  =  <z\ 
that  is,  when  z  =  1.  Hence  this  equation  must  necessarily  be  divisible  by  (z  —  1)*.  Performing  the  divi- 
sion we  find  it  is  so,  and  the  quotient  gives 

«sz4  +  2«5!23  +  3a2z2  +  2  z  -(-  1  =  o;  (y) 

for  determining  the  remaining  values  of  z. 


366  LIGHT. 

Light          As  this  investigation  is  rigorous,  nothing  having  been  omitted  or  neglected  as  small,    *e    have    here    the      Pan  I. 
^-•-V-—'  complete  solution  of   the  problem,  whatever  be  the  aperture  of  the  mirror.      If   this  be    supposed  small  in  ^— - -v— «^ 

157.      comparison  with  the  radius,  an  approximation  to  the  value  of  z  will  be  had  by  t'>e  series  thence  derived, 
Case  when 
tne  aperture  19  9  1395 

is  moderate.  -  =  —  —    a     —  .   a*  —  a    —  &C 

2  32  32  4090 


and  of  course  since  Y  —  a'  :3. 


27  675 

a     ~  °    - 


>04S 


158.  The  first  term  of  this  series  is  sufficient  for  most  cases  which  occur  in  practice,  and  gives 

Case  where 

tne  aperture  as 

is  small  Y  =  --  (a) 

when  com- 
pared to 
radius.          or,  supposing  r  the  radius  of  curvature  of  the  reflector, 

Y—  £         <*> 

The    lateral    aberration    corresponding   to  the  semi-aperture  a  is,  by  the  equation  (j),  Art.  133,  equal  to 
as 
—  5  ;  consequently,  in  the  case  of  small  apertures,  the  radius  of  the  least  circle  of  aberration  is  equal  to  4. 

of  the  lateral  aberration  (at  the  focus)  of  the  exterior  annulus. 

3  3 

i;,g  Carol.  The  least  circle  of  aberration  is  nearer  the  mirror  than  its  principal  focus,  by  —  fgor  —  -   the  lon- 

3  a8 

gitudmal  aberration   =  -   .    -    —  . 
16  r 

JCQ  To  complete  the  theory  of  caustics,  it  only  remains  to  examine  the  degree  of  concentration  of  the  reflected 

Density  of   rays  at  any  assigned  point.     To  this  end,   let  S  (fig.  24)  be  any  point,  and  through  it  let  PS  Y  7  be  drawn 

reflected       touching  the  caustic  in  Y.     Then  S  may  be  regarded  as   lying  in  a  conical  surface  generated  by  the  revo- 

raysatany    lution  of  the  tangent  P  Ysg,  about  the  axis;  and  all  the  rays  in  the  annulus,  generated  by  the  revolution  of 

s~  the    element    PP',  will  be    contained    in    the   hollow    conoidal  solid    formed  by  the  revolution  of  the  figure 

ifg/aJ        PP'Y/9  about  the  same  axis.     Hence  at  S  the  rays  will  be  concentrated:  first,  in  a  plane  parallel  to  that 

of  the  paper,  in  the  ratio  of  PP'  to  S  S',  or  P  Y  to  S  Y  ;   and,  secondly,  in  a  plane  perpendicular  to  that  of 

the  paper,  or  in  the  ratio  of  the  circumferences  of  the  circles  generated  by  the  revolution  of  P  and  of   S, 

that  is,  in  the  ratio  of  these  radii  P  M  :  S  T.     On  both  accounts,  therefore,  the  concentration  at  S   will  be 

PM  PY  PC  PY 

represented  by  -      -    x  ,    or    —  --  x      sy  .      If,  therefore,  we  represent  by  1  the  density  of  the 

rays  immediately  on  their  reflexion  at  P,  their  density  at  S  corresponding,  will  be  represented  by  -  '  —  -, 

S  Y  .  S  q' 
and  this  is  true,  whatever  be  the  situation  of  S. 

161.  But  there  are  now  several  cases  to  be  distinguished.     First,  when  S  is  situated  in  any  part  of  the  spaces 
1st  case.       K  H  V,  N  D  W.  no  such  tangent  can  be  drawn  to  cut  the  reflector  within  its  aperture  A  B  ;  therefore  these 

spaces  receive  no  rays  at  all,  and  the  density  =  o  in  every  point. 

162.  Secondly,  when    S  is   situated   anywhere  within  the  spaces  A  G  B,    V  H  F  E,    E  F  D  W,    only  one    such 
?nd  case,     tangent  can  be  drawn  to  cut  the  reflecting  curve  between  A  and  B.      So  that  in  these    spaces  the  density 

PY.P9 

is  simply  represented  by  D  =  —  -  —  . 

o  i  .  o  (] 

163  Thirdly,  within  the  spaces  KGH  and  M  GD  two  tangents  can  be  drawn  from  any  point  S,  both  touching1 

3rd  case  1ne  branch  FA;  on  the  same  side  of  the  axis  as  the  point  S.  If  we  suppose  P,  Y,  S  <?,  and  P,  Ye  S  q,  to  be 
these  tangents,  S  will  receive  rays  belonging  to  both  these  converging  conoids,  and  the  density  will  therefore 
be  the  sum  of  those  belonging  to  either,  or 

PY,.P?.  PYe.Pft 

SY.-S,,  SYS.S9,' 

Fig.  25.       See  fig.  25. 

164-  Fourthly  and    lastly,  within  the    space  FHGD  there  maybe  drawn  three  tangents  9,  S  Y,  P,,  ^SY'.P,, 

4th  case.      an(j  (?j  g  ya  P,,  all  falling  within  A  B,  the  two  first  (fig.  26)  touching  the  branch  FA  on  the  same  side  as  S,  th«- 


L  I  G  H  T.  367 

third  on  the  opposite  side.     The  former  belong  to  cones  of  rays  converging  to  q,  q^  the  latter  to  a  cone  con-     P»rt  I. 
'  verging  to  qa,  but  intercepted  by  S  after  meeting  at  q,  and  again  diverging.     Hence,  in  this  case,  the  density  will  — — vx-»' 

Fig.  26. 

PY.  .  Pg,  PY8.P9a  PYa.P93 

be  expressed  by  )  =   -gy^^-    +    ~sY77s^    +    -g^Ts^ 

It  would  lead  into  too  great  complication  to  attempt  developing  the  actual  value  of  these  fractions  in  terms  of  APP''catlon 

the  coordinates  of  S,  and  we  will  therefore  merely  apply  them  to  some  remarkable  positions  of  S.  ^  ^"Js" 

Cote  1   S  in  the  axis,  beyond  the  principal  focus,  or  between  the  mirror  and  its  focus  for  extreme  rays  G.    Here       jgr, 

/  P  F  V  Ca!i(;  ' 

Y  coincides  with  F,  and  q  also  does  the  same  therefore  in  this  case,  D  =   I  1 ,    which    shows    that    the 

density  is  inversely  as  the  square  of  the  distance  of  S  from  the  principal  focus. 

Case  2.    S  in  the  axis  between  the  principal  focus  and  the  focus  for  extreme  rays  G,  (z.  K.  in  the  line  GF.)      166. 
Here  S  <j,  =  o,  S  q.t  =  o,  S  93  =  o ;  so  that  here  all  the  three  several  component  portions  of  D  are  infinite,  and  C"86  2 
of  course  the  density  is  infinitely  greater  than  on  the  surface  of  the  reflector. 

Case  3.  S  at  F.  Here  not  only  S  q  =  o,  but  also  S  Y  ;  therefore  at  F  the  density  is  infinitely  greater  than  167 
in  the  last  case,  and  is  the  greatest  which  exists  anywhere.  Case  3. 

Case  4.  S  anywhere  in  the  caustic.  Here  S  Y  =  o,  therefore  in  this  case  also  D  is  infinite,  or  the  density  168. 
infinitely  greater  than  at  the  surface  of  the  reflector;  and  as  S  approaches  F,  this  is  still  further  multiplied  by  Case -4. 
the  diminution  of  all  the  values  of  S  q. 

Case  5.     S  anywhere    in  H  z  D,  the  circle  of  least  aberration.     At  the  centre  z  and  the  circumference  H  the       1G9. 
density  is  infinite.     Between  these  two  positions,  finite,  diminishing  to  a  minimum,  and  again  increasing  accord-  Ca-n-  .1 
ing  to   a  law  too  complicated  to  be  here  investigated.     It  will  be  observed,  that  the  relations  expressed   in 
these  articles  (160—169)  are  general,  and  not  restricted  to  the  case  where  the  reflecting-  surface  is  spherical. 

In  all  the  foregoing  reasoning  the  point  S  is  supposed  to  receive  the  rays  perpendicularly.     The  density  of      170. 
the  rays  therefore  here  intended  must  be  understood  to  mean,  The  number  of  rays  not  incident  on  a  given  par-  Illumination 
ticular  plane  surface,  but  passing  through  a  given  infinitely  small  spherical  portion  of  space,  or  received  upon  of  ascreen 

an  infinitely  small  spherical  body  at  S. 

*.  '  ..     .  the  reflected 

In  cases,  however,  where  the  aperture  is  small,  a  screen  perpendicular  to  the  axis  will  receive  the  rays  from  rayS 

every  point  very  nearly  at  a  perpendicular  incidence ;  and  hence  the  above  expressions  will  in  this  case  represent 
the  intensity  of  illumination  of  the  several  points  in  such  a  surface,  the  screen  being,  however,  supposed  to  stop 
none  of  the  incident  light. 

For  further  information  respecting  caustics,  the  reader  is  referred  to  Tschirniius,  Leipsic  acts  1682,  and  Hist, 
de  I'Acad.,  torn.  ii.  p.  54,  1688  ;  to  De  la  Hire's  Traite'  des  Epicycloides,  and  Mem.  de  I'Acad.,  vol.  x. ;  to 
Smith's  Optics ;  Carre,  Mem.  de  I'Acad.,  1703;  J.  Bernoulli!,  Opera  Omnia,  vol.  iii.  p.  464  ;  I'Hopital  Analyse 
des Infin iment  Petiis ;  Hayes's  Fluxions;  Petit,  Correspondence  de I'Ecole  Poll/technique,  ii.  553;  Malus,  Journal 
de  I'Ecole  Polytech.,  vol.  vi. ;  Gergonne,  Annales  des  Matliematiqucs,  xi.  p.  229  ;  De  la  Rive,  Dissertation  sur 
les  Causticjues,  fyc. ;  Sturm,  Annales  d<:s  Math.,  xvi. ;  Gergonne,  ditto. 


OF  THE   REGULAR   REFRACTION  OF  LIGHT  BY  UNCKYSTALLIZKD  MEDIA. 

§  VI.   Of  the  Refraction  of  Homogeneous  Light  at  Plane  Surfaces. 

When  a  ray  of  light  is   incident  on  the  surface  of  any  transparent  uncrystallized  medium,  a  portion  of  it  is       171. 
reflected  ;  another  portion  is  dispersed  in  all  directions,  and  serves  to  render  the  surface  visible ;  and  the  remainder 
enters  the  medium  and  pursues  its  course  within  it. 

In  the  reflexion  of  light,  the  law  of  reflexion,  as  far  as  regards  the  direction  of  the  reflected  ray,  is  the  \~-i. 
same  for  all  reflecting  media;  the  angle  of  reflexion  being  equal  to  that  of  incidence  for  all.  In  refraction, 
however,  the  case  is  otherwise,  and  each  different  medium  has  its  own  peculiar  law  of  action  on  light ;  some 
turning  a  ray  incident  at  a  given  angle  more  out  of  its  course  than  others.  Whatever  be  the  nature  of  the 
refracting  medium,  the  following  general  laws  are  found  to  hold  good,  and  suffices  (when  the  medium  is  known; 
to  determine  the  direction  of  the  refracted  ray. 

1st.  The  incident  ray,  the  perpendicular  to  the  surface  at  the  point  of  incidence,  and  the  refracted  ray,  all  lie      173 
in  the  same  plane. 

2nd.  The  incident  and  refracted  rays  lie  on  opposite  sides  of  the  perpendicular.  j^ 

3rd.  Whatever  be  the  inclination  of  the  incident  ray  to  the  refracting  surface,  the  sine  of  the  angle  included       175 
between  the  incident  ray  and  the  perpendicular  is  to  the  sine  of  that  included  between  the  refracted  ray  and  the 
perpendicular  in  a  constant  ratio. 

These  laws  equally  hold  good  for  plane  and  for   curved  surfaces,  and  are  found   to  be  verified  with  perfect       1-5 
precision  by  the  most  delicate  experiments,  and  all  the  phenomena  of  refracted  light  to  take  place  in  exact  con- 
formity with  the  results  deduced  from  them  by  mathematical  reasoning. 


368  LIGHT. 

Light          Let  A  C  B  (fig.  23)  be  the  refracting  surface,  P  C  p  the  perpendicular  to  it  at  the  point  of  incidence  C,  S  C      Pttrt 
-  the  incident,  and  C  s  the  refracted  ray.     Then  we  shall  have  v~" "*"" 

sin  P  C  S  :  sin  p  C  *  :  :  fi  :   1, 

/t  being  a  constant  quantity  ;  that  is,  constant  for  the  same  medium  A  B,  though  its  value  is  different  for  different 
media. 

It  is  usual,  for  brevity,  to  speak  of  the  sine  of  incidence,  and  the  sine  of  refraction,  instead  of  the  sines  of  the 
angle  of  incidence,  and  the  angle  of  refraction. 

,_n  m,  .     .  .     sin  of  incidence 

ivy.  Ine  numerical  value  ot  the  quantity  fi,  or  or  -  -  in  any  medium,  must  be  ascertained  before 

sin  of  refraction 

the  law  of  refraction  in  that  medium  can  be  regarded  as  perfectly  known.  This  may  be  done  experimentally 
by  actually  measuring  the  angle  of  refraction  corresponding  to  any  one  given  angle  of  incidence,  for  the  value 
of  the  above  fraction  being  thus  determined  for  one  incidence  holds  equally  for  every  other,  or  by  other  more 
easy  or  more  refined  modes  to  be  described  hereafter.  This  quantity  /*  is  called  the  index  of  refraction  of  the 
IOD'  medium  A  B. 

The  medium  in  which  the  ray  proceeds  previous  to  its  incidence  on  A  B  is  here  regarded  as  a  vacuum.  If 
the  medium  A  B  be  also  a  vacuum,  it  is  clear  that  the  ray  will  not  change  its  course ;  so  that  the  angle  of  inci- 
dence will  equal  that  of  refraction,  and  the  value  of  /i  will  be  equal  to  1.  This  is  the  lowest  value  of  n,  as  no 
medium  has  yet  been  discovered  which  refracts  rays  from  the  perpendicular  when  incident  from  a  vacuum.  The 
greatest  value  of  u  yet  known  is  3,  when  the  refraction  is  made  into  chromate  of  lead ;  and  between  these  limits 
almost  every  intermediate  gradation  has  been  found  to  belong  to  some  one  or  other  transparent  body.  Thus 
for  air  at  its  ordinary  density  fi=  1.00028,  while  for  water  it  is  1.336,  for  ordinary  crown  glass  1.535,  for  flint 
glass  1.60,  for  oil  of  cassia  1.641,  for  diamond  2.487,  and  for  the  greatest  refraction  of  chrom  ite  of  lead  3.0. 
It  is  a  general  law  in  Optics,  that  the  visibility  of  two  points  from  one  another  is  mutual,  whatever  be  the 

Refraction    course  pursued  by  the  rays  which  proceed  from  one  to  the  other.     In   other  words,  that  if  a  ray  of  light  pro 
""  (  ceeding  from  A  arrives   by  any  course  at  B,  however  often  reflected,  refracted,  &c.,  a  ray  can  also  arrive  at  A 

._ from  B  by  retracing  precisely  the  same  course  in  a  contrary  direction.     It  follows  from  this,  that  if  the  ray  S  C 

incident  on  the  exterior  surface  of  a  medium  A  B,  (fig.  23,)  pursue  after  refraction  the  course  Cs,  then  will  a 
ray  sC,  incident  on  the  exterior  surface  of  the  medium,  be  refracted  out  of  it  in  the  direction  CS,  being  bent 
from,  the  perpendicular.  Consequently,  since  in  this  case  the  angle  of  incidence  is  the  same  with  the  angle  of 

refraction  in  the  former  case,  and  vice  vend,  we  shall  have  here  J  —   =  —  .     Thus  we  see  that  the 

sin  refraction  /i 

index  of  refraction  out  of  any  medium  into  vacuum  is  the  reciprocal  of  the  index  of  refraction  into  the  medium 
from  the  vacuum. 

Hence  it  follows,  that  a  ray  can  be  intromitted  into  any  medium  from  a  vacuum  at  any  angle  of  incidence;  for 

since  sin  refr.  =  sin  p  c  s  =  —  .  sin  P  C  S,  the  value  of  n  being  greater  than  1,  the  sine  of  pcs  will  neces- 
sarily be  less  than  that  of  P  C  S,  and  of  course  less  than  unity ;  so  that  the  angle  of  refraction  can  never  become 
imaginary.  Thus,  as  the  angle  of  incidence  PCS  increases  from  o,  or  as  the  ray  S  C  becomes  more  and  more 
oblique  to  the  surface  till  it  barely  grazes, it,  as  at  S''  C,  the  refracted  ray  becomes  also  more  oblique,  but  much 

less  rapidly,  and  never  attains  a  greater  obliquity  than  the  situation  C  /',  in  which  sin  p  C  s"  •=  —  —  =  —  . 

Limit  of  the  This  limiting  angle,  then,  is  the  maximum  angle  of  refraction  from  vacuum  into  the  medium,  and  its  value  in  any 
given  medium  is  found  by  computing  the  angle  whose  sine  is  the  reciprocal  of  the  index  of  refraction.     Thus  in 

water  the  angle  of  refraction  cannot  exceed  arc  sin ; ,  or  48°  27'  40".     In  crown  glass  the  limit  is 

40°  39',  in  flint  38°  41',  in  diamond  23°  42',  while  for  the  greatest  refraction  of  chromate  of  lead  the  limit  is  so 
low  as  19°  28'  20". 

183.  Conversely,  when  a  ray  is  incident  on  the  interior  surface  of  the  medium,  at  any  angle  less  than  the  limiting 

Limit  to  the  . 

possibility     angr]e  whose  sine  is  - — ,  it  will  be  refracted  and  emerge  according  to  the  law  laid  down  in  Art.  181.  being  bent 
of  a  ray  s  ^ 

any  me-      from  the  perpendicular.     But  as  the  angle  of  incidence  pCs  increases,  the  angle  of  refraction  PCS  increases 

ilium.          more  rapidly  ;  and  when  the  former  angle  has  reached  the  limiting  value  p  C  /',  the  transmitted   ray  emerges  in 

the  direction  C  S",  barely  grazing  the  external  surface.     If  the  angle  of  incidence  be  still  further  increased,  the 

angle  of  refraction  becomes  imaginary  ;  for  we  have  sin  P  C  S  =  a  x  sin  pCs,  and  if  sin  p  C  a  7  — ,  the  sine 

When  the    of  P  C  S  must  be  greater  than  unity.     This  shows  that  the  ray  cannot  emerge ;  but  it  does  not  inform  us  whi'.'. 
iy  cannot  j,ecomes  of  ;t_     rpo  ascertain  this,  we  must  have  recourse  to  experiment;   from  which  we  learn,  that  after  this 
reflated  ls  limit  is  passed,  the  ray,  instead  of  being  refracted  out  of  the  medium,  is  turned  back  and  totally  rt'Jlt'.ctcd.  within 
it,  making  the  angle  of  reflexion  p  C  S"'  =  p  C  /". 


LIGHT.  369 

Lignt.          When  the  ray  is  incident  on  the  exterior  surface  of  the  medium,  a  portion  is  reflected  (R)  and  the  remainder      Part  I. 
•v^*'  (r)  refracted.     The  ratio  of  R  to  r  is  smallest  at  a  perpendicular  incidence,  and  increases  regularly  till  the  inci-  **—~\s-~-' 
dence  becomes  90° ;  but  even   at  extreme  obliquities,  and  when  the  incident  ray  just  grazes  the  surface,  the       184. 
reflexion  is  never  total,  or  nearly  total,  a  very  considerable   portion   being  always  intrornitted.      On   the  other  ?hlsreflex" 
hand,  when  the  ray  is  incident  on  the  interior  surface,  the  reflected  portion  (R)  increases  regularly,  with  a  very  10 

moderate  rate  of  increase,  till  the  angle  of  incidence  becomes  equal  to  the  critical  angle,  whose  sine  is —  ;  when 

it  suddenly,  and,  as  it  were,  per  satom,  attains  the  whole  amount  of  the  incident  light,  and  the  refracted  portion 
(r)  becomes  zero.  This  sudden  change  from  the  law  of  refraction  to  that  of  reflexion — this  breach  of  continuity, 
as  it  were,  is  one  of  the  most  curious  and  interesting  phenomena  in  Optics,  and  (as  we  shall  see  hereafter)  is 
connected  with  the  most  important  points  in  the  theory  of  light. 

The  reflexion  thus  obtained,  being  total,  far  surpasses  in  brilliancy  what  can  be  obtained  by  any  other  means  ;       185. 
from  quicksilver,  for  instance,  or  from  the  most  highly  polished  metals.      It  may  be  familiarly  shown  by  filling  a  Experiment 
glass  (a  common  drinking-glass)  with  water,  and  holding  it  above  the  level  of  the  eye,  (as  in  fig.  24,  No.  2.)    If  il'llstraling 
we  then  look  obliquely  upwards  in  the  direction  E  a  c,  we  shall  see  the  whole  surface  shining  like  polished  silver,  reflSex°uan 
with  a  strong  metallic  reflexion  ;    and  any  object,  as  a  spoon,  A  C  B,  for  instance,  immersed  in  it  will  have  its  Fi».24 
immersed  part  C  B  reflected  on  the  surface  as  on  a  mirror,  but  with  a  brightness  far  superior  to  what  any  mirror  No.  2. 
would  afford.     This  property  of  internal   reflexion  is  employed    to  great  advantage  in   the  camera   lucida,  and 
might  be  turned  to  important  uses  in  other  optical  instruments,  especially  in  the  Newtonian  telescope,  to  obviate 
the  loss  of  light  in  the  second  reflexion,  of  which  more  hereafter. 

Some  curious  consequences  follow  from   this,  as  to  vision  under  water.     An  eye   placed  under   perfectly  still       186. 
water    (that  of  a  fish,  or  of  a  diver)  will    see  external   objects  only  through  a  circular  aperture  (as  it  were)  of  Appear- 
96°  55'  20"  in  diameter  overhead.     But  all  objects  down  to  the  horizon  will  be  visible  in  this  space  ;   and  those  j1"0'65.  0{'ex" 
near  the  horizon  much  distorted  and  contracted  in  dimensions,  especially  in  height.     Beyond  the  limits  of  this  :ects  to  a 
circle  will  be  seen  the  bottom  of  the  water,  and  all  subaqueous  objects,  reflected,  and  as  vividly  depicted  as  by  spectator 
direct  vision.     In  addition  to  these  peculiarities,  the  circular  space   above-mentioned    will   appear  surrounded  underwater 
with   a  perpetual  rainbow,   of  faint  but  delicate  colours,  the  cause  of  which  we  shall  take   occasion  to  explain 
further  on.     But  we  need  not  immerse   ourselves  in  water  to   see,  at  least,  a  part  of  these   phenomena.     We 
actually  live  under  an  ocean  of  air,  a  feebly  refracting  medium,  it  is  true,  in  comparison   with  water  ;  and  our 
vision  of  external  objects  near  the  horizon  is  modified  accordingly.     They  are   seen  distorted  from   their  true 
form,  and  contracted  in  their  vertical  dimensions  ;  thus  the  sun  at  setting,  instead  of  appearing  circular,  assumes  Elliptical 
an  elliptical,  or  rather  compressed  figure,  the  lower  half  being  more  flattened  than  the  upper,  and  this  change  form  of 
of  figure  is  considerable  enough  to  be  very  evident  to  even  a  careless  spectator.     The  spherical  form  of  the 
atmosphere,  and  its  decrease  of  density  in  the  higher  regions,  however,  prevent  the  rest  of  the  appearances  above 
described  from  being  seen  in  it. 

If  a  medium  be  bounded  by  parallel  plane  surfaces,  a  ray  refracted  through  it  will  have  its  final   direction       187. 
after  both  refractions  the  same  as  before  entering  the  medium.  Refraction 

Let  A  B,  D  F  be  the  parallel  surfaces  of  the  medium,  and  S  C  E  T  a  ray  refracted  through  it,  P  C  p,  Q  E  q,   thro"Sl1 
perpendiculars  to  the  surfaces  at  C  and  E,  then  we  have 

sin  S  C  P  :  sin  p  C  E  (=  sin  C  E  Q)  :  :  p.  :  1 )  No' 2'°' 

f  and,  compounding  these  proportions, 
sin  CEQ  :  sin  qCT  :  :  1  : /J 

sin  S  C  P  :  sin  q  E  T,  and  therefore  S  C  P  =  q  E  T,  and  the  ray  E  T  is  parallel  to  S  C. 

This  proposition  may  be  proved  experimentally,  by  placing:  the  plane  glass  of  a  sextant  (unsilvered)  before  the  Experimen- 
object-glass  of  a  telescope   directed  to  a  distant  object,  or  before  the  naked  eye,  and  inclining  it  at  any  angle  to  tal  Proof- 
the  visual  ray.     The  apparent  place  of  the  object  will  be  unchanged. 

Experiment.     Let  a  plate  of  glass,  or  any  other  transparent  medium,  be  placed  parallel  to  the  horizon,  and  on       IftS. 
it  let  any  transparent  fluid  be   poured,  so  as  to  form  a  compound   medium   consisting  of  two  media  of  different  Refraction 
refractive  indices,  in  contact,  and  bounded  by  parallel  planes  ;  and  let  an  object  above  this  combination,  a  star,  atthe  com- 
for  instance,  be  viewed  by  an  eye  placed  below  it,  or  through  a  telescope.     It  will  be  found  to  appear  precisely  monsurface 
in  the  same  situation  as  if  the  media  were  removed,  whatever  be  the  altitude  of  the  object,  or  star.     It  follows  medu°in 
from  this,  that  a  ray  S  B  (fig.  26,  No.  2)  incident  on  such  a  combination  of  media,  A  F  and  D  I,  as  described,  contact. 
will  emerge  in  a  direction  H  T  parallel  to  the  incident  ray  S  B.  Fig-  Wi, 

Proposition.     Let  there  be  any  two   media  (No.  1   and  2)  whose   respective  indices  of  refraction  from  a  "**• 
vacuum  into  each  are  p.  and  p!.     Then  if  these  media  are  brought  into  perfect  contact,  (such  as  that  of  a  fluid  ,    '**•?• 
with  a  solid,  or  of  two  fluids  with  one  another,)  the  refraction  from  either  of  them  (No.  1)  into  the  other  (!Vo.  2)  refac°t!,m 

u!  from  one 

will  be  the  same  as  that  from  a  vacuum  into  a  medium,  whose  index  of  refraction  is  — ,   the    index   of  refrac-  medium  int 

f1  another. 

tion  of  the  second  medium  divided  by  that  of  the  first. 

Let  D  E  F  (fig.  26,  No.  2)  be  the  common  surface  of  the  two  media,  and  let  them  be  formed  into  parallel 
plates  A  F,  D  I,  as  in  the  experiment  last  described ;  then  any  ray  S  B  incident  at  any  angle  on  the  surface  A  C 
will  emerge  at  G  I  in  a  direction  H  T  parallel  to  S  B.  Let  B  E  H  be  its  path  within  the  media,  and  draw  the 
perpendiculars  P  B^,  Q  E  q,  II  H  r,  then 

VOL.  iv  3  c 


370  LIGHT. 

Light.  sin  S  B  P  :  sin  E  B  p  =  sin  B  E  Q  :  :  /t  :   1  Put  I. 

sin  R  H  E  =  sin  q  E  H  :  sin  r  H  T  =  sin  P  B  S  :  :  1  :  /t  , 
and,  compounding  these  proportions 

sin  HE,  :  sin  BEQ  ::»:»'; 


'  «in  B  E  Q 


_ 
sin  H  E  9  u 

Absolute  But  B  E  Q  is  the  angle  of  incidence,  and  H  E  q  that  of  refraction,  at  the  common  surface  of  the  media,  con- 

and  relative  sequently  the   relative  index,   or  index   of  refraction  from   the  first   into  the  second,  is  equal  to  the  quotient 

-  of  the  absolute  indices  fif,  fi,  of  the  second  and  first,  or  their  indices  of  refraction  from  vacuum. 

190.  This  demonstration,  it  is  true,  holds  good  only  for  the  case  when  the  angles  of  incidence  and  refraction  at  the 
common  surface  are  both   less  than  the  limits  of  the  angles  of  refraction  from  vacuum  into  each   medium.     If 
they  exceed  these  limits,  the  proposition  however  still  holds  good,  as  may  be  shown  by  direct  measures  of  the 
angles  of  incidence  and  retraction  in  any  proposed  case.     At  present,  therefore,  we  must  receive  it  as  an  experi- 
mental truth. 

191.  Example.  Required  the  ratio  of  the  sine  of  incidence  to  that  of  refraction  out  of  water  into  flint  glass.     The 
refractive  index  of  flint  glass  is  1.60,  and  that  of  water  1.336,  therefore  the  refractive  ratio  required  is 

_L~_    =  M97. 
1.336 

192.  If  the  index  p,  =  —  \,  the  general  law  of  refraction  coincides  with   that  of  reflexion.     Thus  all  the  cases  of 
reflexion,  as  far  as  the  direction  of  the  reflected  ray  is  concerned,  are  included  in  those  of  refraction. 

Of  the  Ordinary  Refraction  of  Light  through  a  System  of  Plane  Surfaces,  and  of  Refraction  through  Prisms. 

193.  Definition.    In  Optics,  any  medium  having  two   plane  surfaces,   through  which   light  may  be  transmitted, 
inclined  to  each  other  at  any  angle,  is  called  a  prism. 

194.  Definition.  The  edge  of  the  prism  is  the  line,  real  or  imaginary,  in  which  the  two  plane  surfaces  meet,  or  would 
meet  if  produced. 

195.  Definition.  The  refracting  angle  of  the  prism  is  the  angle  on  which  its  two  plane  surfaces  are  inclined  to  each 
other. 

196.  Definition.  The  faces  of  a  prism  are  the  two  plane  surfaces. 

197.  Definition.    The  plane  perpendicular  to  both  surfaces,  and  therefore  to  the  edge  of  a  prism,  is  called  the 
principal  section  of  the  prism,  or  of  the  two  surfaces.     This  expression   has  been  used  in  this  general  sense 
already,  under  the  head  of  reflexion. 

To  determine  the  direction  of  a  Ray  after  Refraction  through  any  System  of  Plane  Surfaces. 

198.  Construction.  Since  the  direction  of  the  ray  is  the  same  whether  refracted  at  the  given  surfaces,  or  at  others 
General        parallel  to  them,  conceive  surfaces  parallel  to  the  given  ones,  all  passing  through  one  point,  and  from  this  point, 

[   but  wholly  exterior  to   the  refracting   media,  let  perpendiculars  C  P,  C  P',  C  P",  &c.  be  drawn  to  each  of  the 
tion  through  surfaces,  (fig.  27.)     Let  S  C  be  the  direction  of  the  incident  ray.     Between  C  P  and  C  S'  draw  C  S'  in  the  plane 

any  system  | 

of  plane        S  C  P,  so  that  sin  P  C  S'  =  —  .  sin  P  C  S,  fi  being  the  index  of  refraction  of  the  first  medium  from  the  medium 

surfaces.  /* 

F>s'  27-  in  which  the  ray  originally  moved,  which  we  will  at  present  suppose  a  vacuum,  then  will  S'  C  be  the  direction  of 
the  ray  after  the  first  refraction.  Again,  let  /if  =  the  relative  refractive  index  of  the  second  medium  out  of  the 
first,  or  ft  ft.'  =  its  absolute  refractive  index  from  a  vacuum  ;  draw  C  S"  in  the  plane  S'  C  P'  so  as  to  make 

sin  P'  C  S"  =  —  r  .  sin  P'  C  S',  then  will  S"  C  be  the  direction  of  the  ray  after  the  second  refraction,  and  so  on. 
."' 

199.  General  analysis.    Let  a  =  S  C  P  the  first  angle  of  incidence,  a'  =  S'  C  P7  the  angle  of  incidence  on   the 
second  surface,  I  =  P  C  P'  the  inclination  of  the  two  first  surfaces  to  each  other,  and  putting,  moreover, 

0  =  P  S'  P7  =   the  angle  which  the  planes  of  the  first  and  second  refraction  make  with  each  other. 

•ty-  =   S  P  P'  =   the  angle  made  by  the  plane  of  the  first  refraction  with  the  principal  section  of  the  two  first 

refracting  surfaces. 

0  =  S'  P'  P  =  the  angle  made  by  the  plane  of  the  second  refraction  with  the  same  principal  section. 
p   =  P  C  S'        the  first,  and  />'  =  P'  C  S"  the  second  angle  of  refraction. 
D  =   S  C  S"         the  deviation  after  the  second  refraction. 


LIGHT.  371 

I.ijnt       We  have  (conceiving  S  S' S"  P  P'  to  be  a  portion  of  a  spherical  surface  having  C  for  its  centre)  in  the  spherical     Part 

•—v^-' triangle  S  P  F  given  P  S',  P  P',  and  the  included  angle,  required  S'P'andthe  angles  PS'P',  PP'S';  and,  v— v 

again,   in  the  triangle  S  S'  S"  given  S  S',  S  S"  and  the   angle  S  S'  S",  required  S  S"  the  deviation.     Or,  in 

symbols,  since  p  and  /  are  the  angles  of  refraction  corresponding  to  the  angles  of  incidence  a,  a',  and  the  indices 

of  refraction  /t,  ft', 

-  sin  <*  =  ft  .  sin  p 


(B) 


cos  a!  =  cos  p  .  cos  I  +  sin  p  .  sin  I  .  cos 


sin  a'  =  fif .  sin  p' 


sin  a!  .  sin  0  =  sin  I  .  sin  ty 
sin  of  .  sin  0  =  sin  p  .  sin  ^r 
-  cos  D  =  cos  (<*  —  p)  .  cos  (of  —  p')  —  sin  (<*  —  p)  .  sin  (of  —  />')  .  cos  9. 

From  these  equations,  which,  however,  are  rather  more  involved  than  in  the  case  of  reflexion,  (Art.  99,  200. 
equation  A,)  we  may  determine  in  all  circumstances  the  course  of  the  ray  after  the  second  refraction  ;  and,  in  like 
manner,  as  in  the  case  of  reflexion,  of  any  of  the  eleven  quantities  <*,  a',  p,  p',  /»,  /*',  I,  0,  (p,  fy,  D,  any  five  being 
given  the  remaining  six  may  be  found,  mid  we  may  then  go  on  to  the  next  refraction,  and  so  on  as  far  as  we 
please.  It  is  needless  to  observe,  however,  that,  except  in  particular  cases,  the  complication  of  the  formula 
becomes  exceedingly  embarrassing  when  more  than  two  refractions  are  considered.  Such  is  the  general  analysis 
of  the  problem ;  but  the  importance  of  it  in  optical  researches  requires  an  examination  in  some  detail  of  a  variety 
of  particular  cases. 

Case  1.  When  two  plane  surfaces  only  are  concerned,  at  both  of  which  the  refractions  are  made  in  one  plane,       201. 
viz.  that  of  the  principal  section  of  the  two  planes,  or  of  the  prism  which  they  include.  Case  1. 

Let  the  ray  S  C  (fin-.  28)  be  incident  from   vacuum  on  any  refracting  surface  AC  of  a  prism  CAD,  in  the  When  both 
-  -       '  .  ...  '..     .          -°  .    .  *  .          . .        .     refractions 


refraction  from  the  medium  A  C  D  into  the  medium  A  D  E,  then  will  S"  C  be  parallel  to  the  ray  after  the  second 
refraction  ;  draw,  therefore,  D  E  parallel  to  S''  C,  and  D  E  will  be  the  twice  refracted  ray.  As  in  the  general 
case,  calling  S  C  P,  »  ;  S'  C  P,  p ;  S'  C  P',  «' ;  S"  C  P',  P' ;  and  P  C  P',  I,  &c. 


we  have 
and 


sin  a  =  i*  .  sin  /> ;    a!  =  I  +  (> ;    sin  «'=/*'.  sin  p'\ 
+  D  =  SCS"  =  «_/  +  !;    0  =  o;    0  =  o        )' 


The  first  of  these  equations  gives  p  when  /*  and  a  are  known  ;  the  second  gives  the  value  of  »'  when  p  is  found; 
the  third  gives  />'  when  »'  and  /*'  are  known  ;  and  the  last  exhibits  the  deviation  D. 

The  sign  of  D  is  ambiguous.     If  we  regard  a  deviation  from  the  original  direction  towards  the  thicker  part  of      2Q2. 
the  prism,  or  from  its  edge  as  positive,  which  for  future  use  will  be  most  convenient,  we  must  use  the  lower  sign 

or  take  D  =  p'  —  I  —  a  ;  (b) 

but  if  vice  versd,  then  the  upper  sign  must  be  used.     We  shall  adhere  to  the  former  notation. 

Case  2.  If,  in  case  1,  we  suppose  the  medium  into  which  the  ray  emerges  to  be  the  same  as  that  from  which       203. 

j  Case  2. 

it  originally  entered  the  prism,  (a  vacuum,  for  example,)  we  have  «.'  =  .      This    is    the    case    of   refraction  Both  yefrac 

fi  tions  in  one. 

through  an  ordinary  prism  of  glass,  or  any  transparent  substance.     In  this  case,  I  is  the  refracting  angle  of  the  the  faces  of 
prism,  p  its  refractive  index,  (its  absolute  refractive  index  if  the  prism  be  placed  in  vacuo,  its  relative,  if  in  any  a  prism  in 
other  medium,)  and  the  system  of  equations   representing  the   deviation   and  direction   of  the   refracted  ray  vacuo. 
becomes 

sin  a  =  fi  .  sin  p  ""I 

sin  a'  =   I  +  p 

f  •  (c) 

yiu  p'  =  fi  .  sin  a'    f 

iln  D  =  p'-  a  -  I  J 

Carol.  1.    The  deviation  may  bt  expressed  in  another  form,  which  it  will  be  convenient  hereafter  to  refer  to       204. 
For  we  have 

sin  (I  +  D  4   •»)  =  sin  f  =  P  •  s'n  "'  =  f-  •  sin  (I  +  f) 


=  n  {  sin  p  .  cos  I  +  cos  p  .  sin  I  } 

f  /        I  V  II) 

=  p.  <  sin  p  —  2  sin  p  .   I  sin  —    1   +  2  .  cos  p .  cos  —  .  sm  —  > 


3  c 


372  LIGHT. 

Light.  f          iy  i  l  Par,l 

v^—Y-^,'  because  cos  I  =  1  —  2  I  sin  —  \     and  sin  I  =  2  .  sin  —  .  cos  —  .  v_  -~^~, 

Now  «    sin  p  =  sin  a  by  the  first  of  the  equations  (c),  hence  we  get  (equation  d) 

sin  (I  +  D  +  a)  =  sin  a  f  2  /«.  .  sin  —  .  cos  (  -  -  +  p  )  ;  (d) 

2  \  2  / 

whence,  I  and  a  being  given,  and  p  calculated  from  the  equation  sin  p  =   —  sin  a,  D  is  easily  had. 

205.  Carol.  2.  If  a  =  o,  or  if  the  ray  be  intromitted  perpendicularly  into  the  first  surface,  we  have  also  />  =  o,  anfl 
the  expression  (d)  becomes  simply 

sin  (I  -f  D)  =  ft  .  sin  I  ,  (e) 

sin  (I  +  D) 
whence  also  u,  —  -  V—  j  -  -  ;  (/) 

Thus  we  see  that  if  p.  .  sin  I  7  1,  or  if  I,  the  angle  of  the  prism,  be  greater  than  sin"1  -  ,*  the  critical  angle,  or 

t* 

the  least  angle  of  total  internal  reflexion,  the  deviation  becomes  imaginary,  and  the  ray  cannot  be  transmitted 
at  such  an  incidence. 

206.  Carol.  3.  The  equation  (_/*)  affords  a  direct  method  of  determining  by  experiment  the  refractive  index  of  any 
1st  mode  of  medium   which  can  be  formed  into  a  prism.     We  have  only  to  measure  the  angle  of  the  prism,  and  the  deviation 
determining  of  a  ray  intromitted  perpendicularly  to  one  of  its  faces.     Thus  I  and  D  being  given  by  observation,  n  is  known. 
re'fracti     °  T'''s  's  not>  however,  the  most  convenient  way  ;  a  better  will  soon  appear. 

byexperi-         Definitions.   One  medium  in  Optics  is  said  to  be  denser  or  rarer  than  another,  according  as  a  ray  in  passing 
ment.  from  the  former  into  the  latter  is  bent  towards  or  from  the  perpendicular.     When  we  speak  of  the  refractive 

207.  density  of  a  medium,  we  mean  that  quality  by  which  it  turns  the  ray  more  or  less  from   its  course  towards  the 
perpendicular  (from  a  vacuum,)  and  whose  numerical  measure  is  the  quantity  fa  the  index  of  refraction. 

208.  Proposition.  Given  the  index  of  refraction  of  a  prism,  to  find  the  limit  of  its  refracting  angle,  or  that  which 
Limit  of  the  if  exceeded,  no  ray  can  be  directly  transmitted  through  both  its  faces. 

refracting         This  limit  is  evidently  that  value  of  I  which  just  renders  the  angle  of  refraction  p'  imaginary  for  all  angles  of 

angle  of  a    incidence  on  the  first  surface,  or  for  all  values  of  a,  that  is,  which  renders  in  all  cases 
prism. 

H  .  sin  {I  +  p  }  —  1  positive, 
or,  sin  (1  +  p)  --  positive  ;    that   is,  (since  I  +  p  cannot  exceed  90°)  which  renders  in  all  cases  I  +  p  - 

sin~'  (  —  j  positive.     Now  p  =  sin~'  --  ,  and  consequently  the  value  of  a  least  favourable  to  a  positive 
\  /*  /  /* 

value  of  the  function  under  consideration  is  —  90°,  which  makes  p  =  —  sin  -'  (  --  j,  its  greatest  negative 
value.  Consequently,  in  order  that  no  second  refraction  shall  take  place,  I  must  at  least  be  such  that  I  — 
2  sin  -  '  (  -  )  shall  be  positive  ;  that  is,  I,  the  angle  of  inclination  of  the  faces  of  the  prism  to  each  other, 

Angle  of  a   or  as  it  is  briefly  expressed,  the  angle  of  the  prism,  must  be  at  least  twice  the  maximum  angle  of  internal 
prism.          incidence. 

209.  For  example,  if  fi  =  2,  1  must  be  at  least  60°.     In  this  case  no  ray  can  be  transmitted  directly  through  an 
equilateral  prism  of  the  medium  in  question. 

.jio  Carol.  4.    If  p.  7  1,  or  if  the  prism   be  denser  than  the  surrounding  medium,  /i  .  sin  I  is   7  sin  I  and  sin  ~  ' 

(/t  .  sin  I)  7  I,  so  that  the  value  of  D  (equation  (d),  Art.  204)  is  positive,  or  the  ray  is  bent  towards  the  thicker 
part  of  the  prism,  (see  fig.  29.)  If  fi  £  1,  or  the  prism  be  rarer  than  the  medium,  the  contrary  is  the  case, 
(see  fig.  30.) 

2ii  Problem.    The  same  case  being  supposed,  (that  of  a  prism  in  vacuo,  or  in  a  medium  of  equal   density  on 

Case  of'      both  sides,)  required  to  find  in  what  direction  a  ray  must  be  incident  on  its  first  surface  so  as  to  undergo  the  least 
minimum     possible  deviation. 
deviation.         Since  D  =  p  —  a,  —  I  ;  (c)  Art.  203,  and  by  the  condition  of  the  minimum,  d  D  =  o,  we  must  have 

d  p'  =  d  u. 
Now  the  equations  (c)  give  by  differentiation 

d  a  .  cos  a  •=.  ft  d  p  .  cos  p  ;      d  a'  =  d  p  ;      dp'.  CIH  p   =  fa  d  a  .  cos  <i, 


that  is  dp',  cos  p'  =  p.  d  p  .  cos  0=0.0. 


COS  p 


*  The  reader  will  observe,  that  by  the  expression  sin  ~' is  meant  what  in  most  books  would  be  expressed  by  arc  sin  =  — . 


LIGHT.  373 

COS  a  .  COS  a'  Part  I. 


(1  —  sin  «*)  (1  —  sin  a's)  =  (1  —  sin  />2)  (1  —  sin  p'2) 
in  which,  for  sin  a  and  sin  p'  writing  their  equals,  fi' .  sin  p  and  p  .  sin  a',  we  get 

1  —  ft2 .  sin  />2  1  —  ft5  .  sin  a'  '- 

1  —  sin  />2  1  —  sin  a'4 

which  gives,  on  reduction,  simply  sin  />2  =   sin  «'%   and  therefore  p  =    +  a.',   that  is   I  +  p  =  I  +  a',   or 
a'  —  I  +  a'.     The  upper  sign   is   unsatisfactory,  as  it  would  give  1  =  0.     The  lower  therefore  must  be  taken, 

which  gives  a'  =  — ,  which  satisfies  the  conditions  of  the  question.     We  therefore  have 

of  =  }gl;   p  =  —  i  I ;   sin  a  =  —  ft  .  sin   I  —  j ;  sin  /  =  +  ft .  sin  (  —  I . 

This  state  of  things  is  represented  in  fig.  31,  for  the  case  where  fc    /    I,  or  where  the  prism  is  denser  than   the  Fig.  31. 
surrounding  medium,  and  in  fig.  32,  for  that  in  which  it  is  rarer,  or  ft  /  1 .     In  both  cases,  a,  being  negative,  Fig.  32. 
indicates  that  the  incident  ray  must  fall  on  the  side  of  the  perpendicular  C  P,  from  the  edge  A  of  the  prism  (as 
S  C).     In  both  cases,  the  equations  P  (=  P  C  S')  =  -  1  I  (=  -  J  P  C  P')  and  a'  =  P'  C  S'  =  +  j  P  C  P', 
indicate  that  the  once  refracted  ray  S'  C  D  bisects  the  angle  P  C  P',  and  therefore  that  the  portion  of  it  C  D 
within  the  prism  makes  equal  angles  with  both  its  faces.     In  both  cases,  also,  the  equality  of  the  angles  a  and  // 
(without  reference  to  their  signs)  shows  that  the  incident  and  emergent  rays  make  equal  angles  with  the  faces 
of  the  prism,  and  therefore  that  it  is  of  no  consequence  on  which  face  the  ray  is  first  received. 

Carol.  5.  In  this  case,  also,  we  have  the  actual  amount  of  the  deviation  212. 

Expression 
for  the 

D  =  p'  -  a  -  I  =  2  sin  -  '    U  .  sin   —  j  -   I.  (/) 


minimum 
deviation. 


I  +  D  I 

Hence  also  sin =  /<.  .  sin  —  . 

Carol.  6.  In  the  same  case,  I  being  given  by  direct  measurement,  and  D  by  observation,  of  the  actual  213. 
minimum  deviation  of  a  ray  refracted  through  any  prism,  the  value  of  ft,  its  index  of  refraction,  is  given  at  Another 
once,  for  we  have  deteLanig 

/I   +  l'\  the  index  of 

1    \       g       /  refraction 

ft  =  f .  (g)  of  a  prism 

by  experi- 
ment. 

And  this  affords  the  easiest  and  most  exact  means  of   ascertaining    the  refractive    index    of  any  substance 
capable  of  being  formed  into  a  prism. 

Example.  A  prism  of  silicate  of  lead,  consisting  of  silica  and  oxide  of  lead,  atom  to  atom,  had  its  refracting  214. 
angle  21°  12'.  It  produced  a  deviation  of  24°  46*  at  the  minimum  in  a  ray  of  homogeneous  extreme  red  light :  Examp.e. 
what  was  the  refractive  index  for  that  ray  ? 

I  =  21°  12',  —  =  10°  36',  D  =  24°  46',  — —  =  12°  23' 
2  2 

sin   (-•  +    — —\  =  sin  22°  59'         9.59158 
sin     -|-  =  sin  10°  36'          9.26470 


ft=   2.123  0.32688 

Case  3.    Let  us  now  take  a  somewhat  more  general  case,  viz.  to  find  the  final  direction  and  total  deviation      215. 
of  a  ray,  after  any  number  of  refractions  at  plane  surfaces,  all   the  refractions  being  performed  in  one  plane,  Deviation  or 
and,  of  course,  all  the  common  sections  of  the  surfaces  being  supposed  parallel.  severa^re" 

Supposing  (as  above)  I  to  represent  the  inclination  of  the  first  surface  to  the  second  ;  I'  that  of  the  second  fractions  js 
to  the  third,  &c.  ;    and  I,  I',  &c.  to  be  negative  when  the   surfaces  incline  the  contrary  way  from  one   certain  one  plane. 

side  assumed  as  positive,  taking  also  2,  S',  £",&c 8(«-0  to  represent  the  several  partial  bendings  of  the  rays 

at  the  first,  second,  third,   7ith  surface  respectively,  and  the   •  ther   symbols  remaining  as  before,  we  have  the 
total  deviation,  D  =  S  +  £'  +  ....  &  (n~ ''  .     Now  we  have,  s  nee  in  each  case  9  =:  180°, 


Light 


374 


LIGHT. 


sin  o  =  ft,  .  sin  p  ;    a'  r=  p  +  I  ;     //  .  sin  pf  =  sin  a.'  ;     S  =z 


sin  a'  =  /.  sin  p'  ;  «"  =  p'  +  I'  ;  /'.  sin  p"  =  sin  a"; 
Hence  we  get  (supposing  n  to  represent  the  number  of  surfaces) 


1 
sin  p  =   —  ,  sin  a 


=  a  —  p; 
=  «  —  p' ;  &C.  &C. 


l>»rt  I. 


sin  p'  =  — ;  .  sin  (I  +  p) 
(* 

sin  p"=  — r,  .  sin  (I'  +  p-) 


1 

sin  p  (»-»  =  •          •   .  sin  (I  ("-2>  +  p  (»-2>  ) 

whence  the  series  of  values  p,  p1,  &c.  may  be  continued  to  the  end.     These  determined,  we  get  a,  a',  &c.  by  the 
equations  a  =  o  ;      o'  ==  p  +  I ;     o"  =  p'  +  I' ;    .  .  . .  a  <•"—  '>  =  p  ("— 2>  +  I  (»— V  , 

and  finally  D  =  {  a  -f  a'  + a  (''->>  }  —  {  p  +  /  + p  i»-i)  } 

Now  I  +  I'  +  ...  I  ("-2)  is  the  inclination  of  the  first  to  the  last  surface,  or  the  angle  (A)  of  the  compound 
prism,  formed  of  the  assemblage  of  them  all,  so  that  we  have  in  general 


D  =  a  +  A  -  p  <"- 


(A) 


216.          Let  us  now  inquire,  how  a  ray  must  be  incident  on  such  a  system  of  surfaces  so  that  its  total  deviation  shall 
Case  of       be  a  minimum. 


Since  dD  =  o  and  I,  I',  &c.  are  constant,  we  must  have 


deviation 
after  any 
number  of 
refractions. 


but 


/*  .  sin  p  =  sin  a  ~\         f  ^  dp  .  cos  p  =  d  a .  cos  o 

p/  .  sin  p  =  sin  (p  4-  I)  L  •  <  p'dp1.  cos  p/  =  d  p  .  cos  (p 


&c. 


and  multiplying  all  these  equations  together 


or  simply 


p.  /»'....  f*(B~^.  cos  p  .  cos  p>  .  .  .  .  cos  pt"-1)  =  cos  a  .  cos  a' 


cos 


(t) 


this  equation,  combined  with  the  relations  already  stated,  between  the  successive  values  of  p  and  those  of  a, 
afford  a  solution  of  the  problem  ;  but  the  final  equations  to  which  it  leads  are  of  great  complexity  and  high 
dimensions.  Thus,  in  the  case  of  only  three  refractions,  the  final  equation  in  sin  p  or  sin  p',  &c.  rises  to  the 
sixteenth  degree  ;  and  though  its  form  is  only  that  of  an  equation  of  the  eighth,  yet  there  appears  no  obvious 
substitution  by  which  it  can  be  brought  lower.  The  only  case  where  it  assumes  a  tractable  form  is  that  of  two 
surfaces,  when  the  equation  (;')  which  in  general  may  be  put  under  the  form 

^V*  ----  P-™*  (1  —  sin  ?2)  (1  -  sin  «'"),  &c.  =  (1  —  /ta.  sin  pa)  (1  —  u'    .  sin  p'2),  &c.  (j) 

reduces  itself  by  putting  sin  5  2  =  x,    and  sin  p'a  =  y, 


which,  combined  with  the  equation  p  .  sin  p'  =  sin  (j  +  I) 

or  O*'8  y  +  x  —  sin  Is)2  =  4  ^  *  .  cos  Is  .  x  y, 

gives  a  final  equation  of  a  quadratic  form  for  determining  x  or  y,  and  which  in  the  particular  case  of  /*/*  =  1, 
or  when  the  second  refraction  is  made  into  the  same  medium  in  which  the  ray  originally  moved  before  its  first 
incidence,  gives  the  same  result  we  have  already  found  for  that  case  by  a  similar  process.  Meanwhile,  though 
we  may  not  be  able  to  resolve  the  final  equations  in  the  general  case,  the  equation  (f)  affords  a  criterion  of  the 
state  of  minimum  deviation  which  may  prove  useful  in  a  variety  of  cases. 


LIGHT.  375 

% 

Light.          Case  4.  When  the  planes  of  the  first  and  second  refraction  are  at  right  angles  to  each  other,  required  the  rela       Part  I. 
-— y-^  tions  arising  from  this  condition.  '•— V~~ ^ 

In  this  case  we  have  0  =  90°,  cos  0  r=  0,  sin  0  =  1,  so  that  the  general  equation  (B,  199)  becomes  217. 

Case  when 
sin  a   =  JA.  .  sin  p  \  the  planes 

•  of  the  first 

sin  a  si  ft  ,  sin  p  >  and  cos  a'  =  cos  {  .  cos  1  +  sin  p  .  sin  I  .  cos  ^-.  and  second 

retraction 
Sin  a'  =  sin  I  .  sin  ^/  are  at  right 

angles. 
The  last  of  these  equations,  by  transposition  and  squaring,  becomes 


cos  a'2  —  2  .  cos  a  .  cos  p  .  cos  I  +  cos  £  .  cos  I2  =  sin  p4  .  sin  I*  (1  —  sin  ^r-) 

in  which,  substituting  for  sin  ^  its  value  -       -   deduced  from  the  third  equation,  and  reducing  as  much  as  pos 

sible,  we  obtain 

cos  a'2,  cos  p2  —  2  .  cos  a' .  cos  p  .  cos  I  +  cos  I2  :=  0, 

which,  being  a  complete  square,  gives  simply 

cos  p  .  cos  a'  =  cos  I.  (&) 

This  answers  to  the  equation  cos  a  .  cos  a'  =  cos  I,  obtained,  on  the  same  hypothesis,  in  the  case  of  reflexion 
(104)  ;  for  since  the  latter  case  is  included  in  the  case  of  refraction,  by  putting  ft,  =  —  1  (Art.  192)  we  have 
then  o  =  —  p  and  cos  p  =  cos  «. 

Carol.  \.  If  i  and  f  be  the  inclinations  to  the  first  and  second  surfaces  respectively  of  that  part  of  the  ray       218. 
which  lies  between  the  surfaces,  we  have 

i  =  90°-?        and  f  =  90°  —  a, 

so  that  the  equation  above  found,  gives 

sin  i  .  sin  i'  =  cos  I, 

or  the  product  of  the  sines  of  the  inclination  of  the  ray  between  the  surfaces  to  either  surface  is  equal  to  the 
cosine  of  the  inclination  of  the  two  surfaces.  The  same  relation  may  be  expressed  otherwise,  thus  :  if  we 
suppose  the  ray  to  pass  both  ways  from  within,  out  of  the  prism,  the  product  of  the  cosines  of  its  interior 
incidences  on  the  two  surfaces  is  equal  to  the  cosine  of  their  inclination  to  each  other.  In  this  way  of  stating 
it,  the  case  of  reflexion  is  included. 

Carol.  2.  We  have  also  in  the  present  case  ojo 


1 

sin  p  =±  .  sin  a  ; 

v  /JL  *  —  sin  a 


.    /»»  .  sin  I2  —  sin  a4  1   .    /  ^  .  sin  P  —  sin  a4 

a  =  V  : * — : — « —  ;       sm  c  =  — V  - — i — - — * — 

V  <t «  —  sin  a4  H.      r  u4  —  sin  a 4 


and  cos  D  =  cos  (a  —  ?)  .  cos  (a1  —  p') 

so  that  a  being  given,  all  the  rest  become  known.     The  last  equation  corresponds  to  the  equation  cos  D  =  cos 
2  a  .  cos  2  a  in  the  case  of  reflexion. 


§  VII.     Of  Ordinary  Refraction  at  Curved  Surfaces,  and  of  Diacauttics,  or  Caustics  by  Refraction. 

The  refraction  at  a  curved  surface  being  the  same  as  at  a  plane,  a  tangent  at  the  point  of  incidence,  if  we       220. 
know  the  nature  of  the  surface,  we  may  investigate,  by  the  rules  of  refraction  at  plane  surfaces,  combined  with 
the  relations  expressed  by  the  equation  of  the  surface,  in  all  cases,  the  course  of  the  refracted  ray.     We  shall 
confine  ourselves  to  the  simple  case  of  a  surface  of  revolution,  having  the  radiant  point  in  the  axis. 

Proposition.  Given  a  radiant  point  in  the  axis  of  any  refracting  surface  of  revolution,  required  the  focus  of       221. 
any  annulus  of  the  surface.  General  in- 
Let  C  P  be  the  curve,  Q  the  radiant  point,  Q  q  N  the  axis,  P  M  any  ordinate,  P  N  a  normal,  and  P  q  or  q  P  vestigation 
the  direction  of  the  refracted  ray,  and  therefore  q  the  focus  of  the  annulus  described  by  the  revolution  of  P.  of  the  focus 
Then  if  we  put  ft  for  the  refractive  index,  and,  assuming  Q  for  the  origin  of  the  coordinates,  put  Q  M  =:  x,  guj^e'of ' 
i                                                                                                                                                                     revolution. 

M  P  =  y,  r  =  Vx*  +  y\   p  =  — *-,  we  have  Fie-  3a- 

'/     ' 

sin  QPM  =  — ;  c 

r 


376  LIGHT. 


V  1  +  p* 

consequently  sin  N  P  Q  =  sin  Q  P  M  .  cos  N  P  M  +  sin  N  P  M  .  cos  Q  P  M 

_        x+py_^ 
r,  J\  .,-p*' 

|  y   _|_     p  y 

and  therefore  sin  N  P  q  = .  sin  N  P  Q  =  ; 

g  

and  cosNPg=. ,    if  we  put  Z  =:   J  ^  rt  (i  +  p*)  —  (x  +  py)  •;  (a) 

^  r  v  1  +  p* 

consequently  since  M  P  q  =  N  P  q  +  N  P  M,          we  get 


-  p  (x 

sinMPg  =  —     ,1    .       ^     »         andcosMPg  =  -  -  --  TT~, 
pr  (1  +  p*)  f*r(l  + 

sin  M  P  g 
tan  MPg=  -        *      = 


=   -     —  r-  .    ,    ^. 

cos  M  P  g  -  p  (#  +  p  y)  +  Z 

y       Z 


whence 

Now  we  have  an.  q  =  r  1*1 .  tan  i»i  r  q  =  y  .  lan  in  r-  </  =  = —  — - —     — r , 

*  —  p  (x  +  p  y) 

consequently  Q  9  =  *  +  y  •  tan  M  P  q  =  (z  +  py)  .  -  -„  .  (c) 


222.  Carol.  1.  If  we  put  *  for  the  arc  C  P  of  the  curve,  we  have,  since  rdr  —  xdx  +  ydy  =  dx(x+  py), 


ds    V        /rdr  .  /  ds  V       /  dr 

z= 


5323.          Carol.  2.    If  ft.  =  —  1,  in  which  case  the  refraction  becomes  a  reflexion,  we  have 

Z  =  V  r2  (1  +  p2)  —  (r  -f  py)2  =  y  —  p  x,  writing  for  r4  its  value  x*  +  ya  ;  so  that  the  general  value  above 
found  for  Q  q  reduces  itself  to 

0          „       (x  +  py~)  (px~  y) 

•2px-y(\  -  p*)    ' 
which  is  the  same  as  that  found  in  (6)  Art.  109,  in  the  case  of  reflexion. 

224.          Carol.  3.  If  we  put  P  =  tan  M  q  P  =  cotan  M  P  q  =         M  p-. 


v      -  +  z 

we  have  P 


*  +  py+pZ 

and  the  equation  of  the  refracted  ray,  if  X  and  Y  be  its  coordinates,  (Q  being  their  origin)  will  be  (since  Y  lies 
on  the  opposite  side  of  the  curve  from  Q) 

Y  -  y  =  -  P  .  (X  -  x)  (/) 

225  In  the  case  of  parallel  rays  these  expressions  become  (by  putting  first  x  +  a  for  *,  and  then  making  a  infinite) 

Z=  a  ^  f  (1  +  p4)  —  1 

P  as  — 


A  9  =  x  4-  y  . 


LIGHT.  377 

VIII.    Of  Caustics  by  Refraction,  or  Diacaustics. 


Light.  Part 


The  theory  of  Diacausti  ;s  is  in  all  respects  analogous  to  that  of  Catacaustics  already  explained.     To  find  the      226. 
coordinates  X  and  Y  of  the  point  in  the  diacaustic  which  corresponds  to  the  point  P  in  the  refracting  curve,  we  F'S-  34- 
have  only  to  regard  the  equation  (/)  and  its  differential  with  respect  to  x,  y,  and  p  alone,  as  subsisting  together, 
and  we  get  the  necessary  equations  for  determining  X  and  Y  in  terms  of  x,  y,  as   in  the  case  of  reflexion,  and 

these  are  X  =  x  +  •  Prf+p??      dx;        Y  =  y  -  P  .  —  j^-  d  x  ;  (i) 

the  only  difference  is  in  the  signs  and  in  the  value  of  P,  which,  instead  of  the  formula  (c,  Art.  1  10,)  is  here 
expressed  by  the  more  complicated  function  (e,  Art.  223,)  and  the  equation  of  the  diacaustic  will  be  obtained  as 
before  by  eliminating  all  but  X,  Y  from  these. 

P  +  p 
It  is,  evident,  moreover,  that  if  we  suppose,  as  in  the  theory  of  Cataeawtics,  M  =  —  -—  =j  —  d  x  ;    and  put      227. 

S  for  the  length  of  the  caustic,  and  /  for  the  line  P  y,  we  shall  have,  exactly  as  in  that  theory, 


anddS  =  df+  dx. 


.  .  . 

oX  V  l  +  p: 

See  Art.  139,  143,  144. 

Now  we  have,  substituting  for  P  its  value  (e), 


_         _  . 


_ 
x+py+pZ'  x  +  py  +  pZ 

and  consequently  the  value  of  d  S  becomes 

*    -df+  _!!!,  because  (*  +  py)dx  = 


and  integrating  S  =  const  +  /  +  ; 

V- 
so  that  we  have,  finally,  (fig.  34,)  Fig.  34. 

arc  Fy=  (CF-Py)  +  —  (Q  C  -  Q  P).  (0 

ft 

In  the  case  of  reflexion,  ft  =  —  1,  but  at  the  same  time  the  sign  of  f  is  negative,  because  in  this  case       228 
the  reflected  ray  lies  on  the  same  side  of  the  point  of  incidence  with  the  incident  one  ;  thus  both  terms  of 
the  formula  change  their  sign,  and  this  expression  coincides  with  that  found  Art.  144. 

In    the    case  of  parallel  rays,  we  must   use    the  value  of  P  found    in    Art.  225,  equation  (§•).      Putting      229. 

dp 

q  =  ,  and  executing  the  operations,  we  find,  then, 

d  x 

v  _          1         /*2  (l  +  p*)  -  1 

A  —  X  — 


_ 

P  .  , 

(m) 


Carol.    If  we  suppose  p  =  uo  ,  or  the  refractive  power  infinite,  the   refracted    ray  will    coincide    with  the      230. 
normal,  and  the  caustic  will  be  identical  with  the  evolute  ;  and  it  is  evident  that  the  expressions  (m),  when 
f4  =  co  ,  resolve  themselves  into  the  well-known  values  of  the  coordinates  of  the  evolute. 

If  the  rays  incident  on  the  refracting  curve  do  not  diverge  from  one  point,  but  be  all  tangents  to  a  curve      33  j 
V  V  V",  (fig.  35,)  we  must  put  x  —  a  for  x  in  the  value  of  P,  (eq.  (e)  Art.  224  ;)  and  fix  the  origin  of  the  Fig.  35.' 
coordinates  at  A,  putting  A  Q  =  a  ;  and  if,  then,  we  regard  a  as  variable  according  to  any  given  law,  (or 
regard  x  —  a  at  once  as  a   given  function   of  x,)  and  take  the    differential  of  P  on  this    supposition,  the 
equations  (i)  still  hold  good,  and  suffice  to  define  the  caustic. 

Problem.    The  radiant  point  and  refractive  index  of  a    medium  being   given,  to  determine  the    nature   of      goo 
the  curve  surface  which  shall  refract  all  the  rays  to  one  point. 

Here  we  are  required  to  find  the  relation  between  x  and  y,  so  as  to  make  Q  q  invariable.     Let  Q  q  =  c, 
and  we  have 

77  X  —  V  —  Z 


t 

_  Z  =  V?  (**  +  y»)  (1  +  p") 

This  equation  gives 

(x  +  py)  (p  (x  -  c)  —  y)  =  Z  (x  —  c  +  p  y). 

VOL.  IV.  3  D 


378  LIGHT. 

Light      Squaring1  both  sides,  and  substituting  for  Z  its  value,  we  get  Part  I. 


which,  on  executing  the  operations  indicated  on  the  left  hand  side,  becomes  totally  divisible  by  1  +  p  *,  and 
reduces  itself  to 

(3/2  +  (*-c)2)  =fc*  (x-c 


d  11 

that  is,  putting  for  p  its  value  ,  multiplying  by  d  x2,  and  extracting  the  square  root, 

u  X 

xdx  +  y  dy  (  x  —  c)  d  x  +  y  d  y 

V  x*  +  y*  A/  (x  —  c)2  +  y* 

and  integrating  (each  side  being  a  complete  differential) 

+  y*=:  b  +  ft  .  V(x  —  cy+y  *,  (n) 


which  is  the  equation  of  the  curve  required,  and  belongs,  generally,  to  a  curve  of  the  fourth  order. 
233  Carol.  1.  About  Q,  (fig.  36,)  n  ith  any  radius  Q  A,  arbitrarily  assumed,  describe  a  circle  A  B  D  E,  then  if  C  P 

fig.  36.       be  the  refracting  curve,  and  we  put  Q  A  =  6,  we   have  QP=-v,re-fys,   P  </  =   ^  (x  —  c)a  +  y2,  and  the 
nature  of  the  curve  is  expressed  by  the  property 


B  P  =  ft  .  P  q,          or,  B  P  :  P  q  :  :  p  : :  1. 

e  Q  P 

y"-} 


234.  Carol.  2.  If  b  =  o,  or  the  circle  AB  E  be  infinitely  small,  we  have  Q  P  :  P  q  :  :  p  :  1,  which  is  a  well  known 

property  of  the  circle.     In  fact,  in  this  case  we  have  simply 


In  this,  if  we  change  the  origin  of  the  coordinates  by  writing  x  +    c  for  x  we  find 

fx-  —    1 


The  radius  of  the  circle,  therefore,  is  equal  to  — —   x   Q  q,  and  the  distance  of  its  centre  from  the  radiant 

fig  37.       l'oint  is  — t  _        x  Q  9-     Take  therefore  any  circle  H  P  C  whose  centre  is  E,  (fig.  37,)  and  two  points  Q,  q, 

such  that  Q  E  =  ft  x  E  C  and  Q  C  :  C  q  :  :  ft  :  1.     Then  if  rays  diverge  from  Q,  and  fall  on  the  surface  P  H 
beyond  the  centre,  they  will,  after  refraction  into  the  medium  M,  all  diverge  from  q. 

2  >5.  Carol.  3.  If  ft  =:  —  1,  the  equation  («),  when  freed  from  radicals,  is  only  of  the  second  degree  between  x  and 

y,  and  therefore  belongs  to  a  conic  section.     On  executing  the  reduction  we  get 


which  shows  that  the  radiant  point  Q  is  in  one  focus  and  q  in  the  oUrer,  which  is  the  same  result  as  that  before 

found  by  a  different  mode  of  integration. 

236.  Carol.  4.  When  Q  is  infinitely  distant,  and  the  rays  are  parallel,  we  Trust  shift  the  origin  of  the  coordinates 

For  parallel  from  Q  to  q,  by  putting  c  —  x  for  x,  and  afterwards  supposing  c  infinite.     This  gives 
rays  the 


curve  is  a  V  n«  _  2cx  +  x^+y^  =  b+ft  vx*  +  y 2. 

co»ic 

section.^       Developing  the  first  term  in  a  descending  series,  we  find 

,r*  +  y*  /— 5 2 

Let  c— 6  =  h,  which,  since  6  is  arbitrary,  is  equally  general,  and  may  represent  any  finite  quantity,  then,  as  c 
increases  and  at  length  becomes  infinite,  this  equation  becomes  ultimately 


h  —  x  —  p  */  x*  +  y*. 

Let  C  P  be  a  conic  section,  q  its  focus,  and  A  B  its  directrix,  q  M  ±=  *,  and  P  M  =  y,  then  will  Q  P  =  h  —  x 
if  we  take  q  A  =  h,  and  the  above  equation  we  see  expresses  that  well  known  property  of  a  conic  section,  in 
virtue  of  which  QP  :  Pq  in  a  constant  ratio,  (/»  :  1.) 

237.  Carol.  5.  The  curve  is  an  ellipse  when  Q  P  7  P  q,  or  when  the  ray  is  incident  from  a  rarer  on   a  denser 

medium,  and  an  hyperbola  in  the  contrary  case.     If  Q  P  =  P  q,  the  curve  is  a  parabola  ;  in  this  case  /*  =  1,  and 
the  rays  converge  to  the  focus  at  an  infinite  distance,  i.  e.  remain  parallel. 

To  take  a  single  example  of  the  investigation  of  the  diacaustic  curve,  fiom  the  general  expressions  above 


LIGHT.  379 

Light,      delivered, — let  the  refracting:  surface  be  a  plane,  and  we  shall  have,  fixing  the  origin  of  the  coordinates  at  the      P«rt  I. 
~v-"~*'  radiant  point,  and  supposing  the  axis  of  the  x  perpendicular  to  the  refracting  plane  A  C  B,  ^—-y— -^ 

Caustic  of  a 

x  =  constant  =  Q  C  =  «,      p-  -JL  =  co .         Thus  we  get  &; 

-  « 

surface. 


V  (ft*  -  1)  y*  + 


and  therefore  by  the  equations  (i)  we  get,  substituting  these  values,  40. 


1 


Eliminating  y  from  these,  we  have  the  equation  of  the  caustic 


vi-/»*    jr  y  _  t 

/»  a  ) 

This  is  the  equation  of  the  evolute  of  a  conic  section  whose  centre  is  C,  and  focus  the  radiant  point  Q.  If  ft 
be  greater  than  unity,  or  the  refraction  be  made  into  a  rarer  medium  from  a  denser,  the  conic  section  is  an 
ellipse,  (see  fig.  39,)  and  in  the  contrary  case  an  hyperbola,  (fig.  40.) 

§  IX     Of  the  Foci  of  Spherical  Surfaces  for  Central  Rays. 

Definitions.  The  curvature  of  any  spherical  surface  is  the  reciprocal  of  its  radius,  or  a  fraction  whose  nume-       239. 
rator  is  unity,  and  denominator  the  number  of  units  of   any  scale  of  linear  measure  to  which  the  radius  is  Curvature 
equal.  defined- 

The  proximity  of  one  point  to  another  is  the  reciprocal  of  their  mutual  distance,  or  the  quotient  of  unity  by      240. 
the  number  of  units  of  linear  measure  in  that  distance.  Proximity. 

The  focal  distance  of  a  spherical  surface  is  the  distance  from  the  vertex,  of  the  point  to  which  rays  converge,       241. 
or  from  which  rays  diverge  after  refraction  or  reflexion.  Focal 

The  principal  focal  distance,  or  focal  length,  is  the  distance  from  the  vertex  of  the  point  to  which  parallel      A^' 
and  central  rays  converge,  or  from  which  they  diverge  after  refraction  or  reflexion.  Focal  ln»th 

The  power  of  a  surface  is  the  reciprocal  of  its  principal  focal  distance,  or  focal  length,  estimated  as  in  the      343"° 
definitions  of  curvature  and  proximity.  Power. 

Problem.  To  find  the  focus  of  a  spherical  refracting  surface  after  one  refraction,  for  central  rays.  344. 

Here,  putting  a  for  the  distance  of  the  focus  of  incident  rays  Q,  (fig.  41,)  from  the  centre  E,  we  have  General  ex- 

pressions  for 
the  focal 
'  -     '   f  j/2'  <*•    i    r  y  —  «•  > 

any  annulus 

and  these  substituted  in  the  general  expressions  Art.  221,  give  of  a  sphe- 

rical refract- 
Z  —  V  ,j2  j.2  $1  _)_  (ij2  r2  av\  yi         N  ing  surface 


(a-  x)*  +  y*  =  r*;  P=~ — ;  1+P°~  =  -^-;  x  +  py=a; 

%  distance  of 


y 

Qq~a\l~a(a-x)-yZl  \.          (a) 


These  values  of  Q  q  and  C  q  contain  the  rigorous  solution  of  the  problem,  whatever  be  the  amplitude  (y)  of  the  Focus  for 
annulus  whose  focus  q  is,  and  we  shall  accordingly  again  have  recourse  to  them.  At  present,  however,  our  central  ra 
concern  being  only  with  central  rays,  we  must  put  y  =.  0,  when  we  find  x  =  a  —  r  ;  yZ  =  firx=pr(a~r') 


} 


Carol.  1  .  This  latter  is  the  focal  distance  for  central  rays.     Now,  since  a  —  r  =  Q  C,  this  gives  the  following      245. 
proportion, 

p  .  Q  C  -  Q  E  :  /.  .  Q  C  :  :  C  E  :  C  q.  (c) 

3D2 


380  LIGHT. 


C  F  =  -^  ;     that  is,  C  E  :  C  F  :  :  ft  -  I  :  p.  j 
CE  :EF:  :  p  -  I  :  1,      and  CF  :  FE  :  :  ft  :  1   ) 


Light.          Carol.  2.  If  we  suppose  the  focus  of  incident  rays  infinitely  distant,  or  a  —  as,  and  take  F  the  place  of  q  for     Part  I. 
'^—-v^'— '  central  rays,  on  that  supposition,  F  will  be  the  principal  focus,  and  we  shall  have 

246. 
Focus  for 
parallel  rays 

whence  we  also  find 


247.          These  results  will  be  expressed  more  conveniently  for  our  future  reference  by  adopting  a  different  notation. 
Let,  then, 

R  = =  curvature  of  the  surface,  and  let  positive  values  of  r  and  R  correspond  to  the  case 

where  the  centre  E  lies  to  the  right  of  the  vertex  C,  or  in  the  direction  in  which  the 
rays  proceed. 

D  =  •  (fig.  42)  =  proximity  of  the  focus  of  incident  rays  to  the  surface,  D  being  regarded  as 

positive  when  Q  lies  to  the  right  of  C,  as  in  fig.  42,  and  as  negative  when  to  the  left, 
as  in  fig.  41.  Then,  since  Q  E  =  a,  and  since  in  the  foregoing  analysis  a  is  regarded 
as  positive  when  Q  is  to  the  left  of  E,  we  must  have  (fig.  42)  Q  E  =  —  a,  and 
QC=QE-rEC=r  —  a,  so  that 

D  =  ;          a  =  — — -=—  .     Let  also  m  =  : 

r  —  a  R          D  ft 

F  =  =  power  of  the  surface  : 

/  =  — - —   =  proximity  of  the  focus  of  refracted  rays  to  the  surface. 

Positive  values  of  F  and  /  as  well  as  of  D  and  R,  being  supposed  to  indicate  situations  of  the  points  F,  f,  Q,  E, 
respectively,  to  the  right  of  C,  or  in  the  direction  towards  which  the  light  travels.  This  is,  in  fact,  assuming  for 
our  positive  case  that  of  converging  rays  incident  on  ^a  convex  surface  of  a  denser  medium.  We  shall  have,  then, 

_L        r'_a=_L.  J_       JL  J_ 

Fundamen-  But  equation  (6)  gives  — —  =  — £-- — ^-,  and  substituting  we  shall  get 

tal  equation  *->  q               ftr  (r  —  a) 

for  the  foci 

of  central  /=  (1  —  m)  R  +  m  D.                    (e) 

This  equation   comprises  the  whole  doctrine  of  the  foci  of  spherical  surfaces  for  central  rays,  and  may  be 
regarded  as  the  fundamental  equation  in  their  theory. 

In  the  case  of  parallel  rays,  we  have  D  =  0,  whether  the  rays  be  incident  from  left  to  right,  or  from  right  to 
General  ex-  ]eft  jn  g^er  case>  tj,erl)  y  has  the  same  value,  viz.  (1  —  m)  .  R,  and  the  principal  focal  distance  F  in  either 
pression  lor  *j.i_  i_  •  •  i_  J.L.  A* 

The  power    case  ls  'ne  same>  being  given  by  the  equation 

of  a«v  F  =  (1  -  in)  .  R,  (/) 

which  shows,  moreover,  that  the  power  of  any  spherical  surface  is  in  the  direct  ratio  of  its  curvature. 

Hence  also  we  have  /=  F  +  m  D.  (g) 

250  In  the  case  of  reflexion,  where  ft  =:  —  1,  or  m  =  —  1,  these  equations  become  respectively 

Vundamen-  F  =  2R;          /=  2  R  -  D ;          /=  F  -  D.                    (A) 

tal  expres- 
sions for  the  Such  are  the  expressions  for  the  central  foci  in  the  case  of  a  single  surface.     Let  us  now  consider  that  of  any 
Incase  oT  system  of  spherical  surfaces, 
reflexion"  Problem.  To  find  the  central  fonts  of  any  system  of  spherical  surfaces. 

251.  Let  C',  C",  C'",  &c.  be  the  surfaces.     Q'  the  focus  of  rays  incident  on  C',  Q''  that  of  refracted  rays,  or  the 

Central  focus  of  rays  incident  on  C",  and  so  on.     Call  also  R',  R",  &c.  the  radii  of  the  first,  second,  &c.  surfaces  «',  ft", 

locus  of  a  .     . 

spherical      ^-C-  t'le'r  refract've  indices,  or  —       — j-  into  each  medium  from  that  immediately  preceding,  m'  —  — — ,  m''=.  — , 
surfaces  in- 

Fig.'6^'    &c.     Also  let  D'  =  -^  ,  D"  =  — ^  &c.  and  moreover  let  C'C"  =  t\  C"  C'"  =  t",  &c.  f,  t",  &c.  being 

o  y  ^  \j 

regarded  as  positive  when  C",  C"',  &c.  respectively  lie  to  the  right  of  C',  C'',  &c.  or  in  the  direction  in  which 
the  light  travels  ;  and  if  we  put  g  *  •  =  /,  c*,  =  /",  &c.  F'  =  (1  -  m')  R',  F"  =  (1  -  m")  R",  &c. 

we  shall  have  by  (249) 

f  =  F  +  m'  D' ;        /'  =  F"  +  m"  D",  &c. ;  (i) 


L  I  G  H  T.  381 

Light,     but  we  have  also  Part 

C'  Q'  =  ^r  ;        C"  Q"  =  -~  =  C'  Q"  -  C'  C",  =  -I   -  f  .- 

and  so  on  ;  so  that  we  have,  besides,  the  following  relations, 

D'=D';         D"=    -7,:         PW= 


and  substituting  these  values  of  D",  D'",  &c.  in  the  equations  (z),  and  in  each  subsequent  one,  introducing  the 
values  of/',  /",  &c.  obtained  from  those  preceding,  we  shall  obtain  explicit  values  of/',  /",  &c.  to  the  end. 

The  systems  of  equations  (i)  and  (.;')  contain  the  general  solution  of  the  problem,  whatever  be  the  intervals      252 
between  the   surfaces.     On  executing  the  operations,   however,  for  general  values  of  V,   t",  &c.  the  resulting 
expressions  are  found  to  become  exceedingly  complex,  nor  is  there  any  way  of  simplifying  them,  the  complication 
being  in  the  subject,  not  in  the  method  of  treating  it.     For  further  information  on  this  point,  consult  Lagrange, 
(Sur  la  ThArie  des  Lunettes,  Berlin,  Acad.  1778.)     We  shall  here  only  examine  the  principal  cases. 

Problem.   To  find  the  focal  distance  of  any  system  of  spherical  surfaces  placed  close  together.  253. 

Here  f,  if'.  &c.  all  vanish,  and  the  equations  (f)  and  (j)  become  simply  Foci  of  a 

system  of 
D'  =  D' ;         D"  =  /' ;         D'"  =  /",  &c. ;  spherical 

fi  —  F"  +  m'  D' ;         f"  =  F"  +  m"  D'',  &c. ;  surfaces 

placed  close 
whence  by  substitution  we  obtain  together. 

/"  =  F"  +  m"  F   +  m  m"  D' 

ft>  —  -pi"  +  m'H  f  +  m"'  m"  p"  +  m'"  m"  m'  D', 

which  it  is  easy  to  continue  as  far  as  we  please. 

Carol.  1.    Let  the  number  of  surfaces  be  n,  and  let  M'  represent  /,  or  the  absolute  refractive  index  out  of      254. 
vacuum  into  the  first  medium;    M"  =  /*V,  or  the  absolute  refractive  index  from  vacuum  into  the  second 
medium,  and  so  on ;  /i',  /*'',  &c.  representing  only  the  relative  refractive  indices  from  each  medium  into  that 
succeeding  it.     Thus  we  shall  have 

M  (n)  f(n)  _  Di  +  M1  F'  +  M"  F"  + M  <">  F  w  .  (A) 

Cor.  2.  For  parallel  rays,  in  whichever  direction  incident,  we  have  D'  =  0  ;  and  the  principal  focal  length  of      255. 
the  system,  which  we  will  call  TT;,,  is  given  by  the  equation 

M  <"'  0  w  =  M1  F1  +  M"  F"  + M  W  F <"> .  (0 

Cor.  3.  Hence  it  appears  that  0<n>,  the  power  of  the  system,  or  its  reciprocal  focal  length  for  parallel  rays,  being      256. 
found  by  the  last  equation,  the  focus  for  any  converging  or  diverging  rays  is  had  at  once  by  the  equation 

MW/W  =  MW  0(»)  -f.  D'. 

For  brevity  and  convenience,  let  us,  however,  modify  our  notation  as  follows:  confining  the  accented  letters      257. 
to  the  several  individual  surfaces  of  which  the  system  consists,  let  the  unaccented  ones  be  conceived  to  relate  Fundamen- 
to  their  combined  action  as  a  system.      Thus,  F',  F",  ....   F<">  representing  the  individual  powers  of  the  '*'  exPres- 
respective  surfaces ;   let  F,  without  an  accent,  denote  the  resulting  power  of  the  system.     In   this   view  D'  may  e'e0nntral°foci 
be  used  indifferently ;  accented,  as  relating  to  the  incidence  on  the  first  surface ;   or  unaccented,  as  expressing  Of  an,, 
the   proximity  of  the  focus  of  incident  rays  to  the  vertex  of  the  whole  system.     Similarly,  M  (n)  may  be  used  system  of 
without  an  accent,  if  we  regard  the  total  refractive  index  of  the  system  as  that  of  a  ray  passing  at  one  refraction  spherical 
into  the  last  medium.     This  supposed,  the  equations  (k)  and  (I)  become 

MF  =  M'F'  +  M"F"  + M«  F«  ;  (rn) 

M  /  =  M  F    +  D  ;       M(F  — /)  +  D  =  0.  (n) 

If  the  whole  system  be  placed  in  vacuo,  or  if  the  last  refraction  be  made  into  vacuum,  we  have  M  =  1  =  M  (*•',      2&b. 
and  the  equations  become 

F  =  M'  F'  +  M"  F"  +  . .    .  M  (' 


/=F+D 


)  ,, 

f 


Definitions.  A  lens  in  Optics  is  a  portion  of  a  refracting  medium  included  between  two  surfaces  of  revolution      259 
whose  axes  coincide.     If  the  surfaces  do  not  meet,  and  therefore  do  not  include  space,  an  additional  boundary  is  Le"ses  (ie- 
required,  and  this  is  a  cylindrical  surface,  having  its  axis  coincident  with  that  of  the  surfaces.  tln^Uhed'5" 

The  axis  of  the  lens  is  the  common  axis  of  all  the  bounding  surfaces.  inlo"™'  c  . 

Lenses  are  distinguished  (after  the  nature  of  their  surfaces)  into  double-convex,  with  both  surfaces  convex, 
(fig.  44  ;)  plano-convex,  with  one  surface  plane,  the  other  convex,  (fig.  45  ;)  concavo-convex,  (fig.  46  ;)  double- 
I'uncave,  (fig.  47  ;)  plano-concave,  (fig.  48  ;)  and  meniscus,  (rtg.  49,)  in  which  the  concave  surface  is  less  curved 
than  the  convex.  Also  into  spherical:,  (when  the  surfaces  are  segments  of  spheres  ;)  conoidal,  when  portions 
of  ellipsoids,  hyperboloids,  &o 


382  LIGHT. 

light.          These  different  species  are  distinguished,  algebraically,  by  the  equations  of  the  surfaces,  and  by  the  signs  of    Part  I. 
-— . -v— '  their  radii  of  curvature.     In  the  case  of  spherical  lenses,  to  which  our  attention  will  be  chiefly  confined,  if  we  — ^-« 

260.  suppose  a  positive  value  of  the  radius  of  curvature  to  correspond  to  a  surface  whose  convexity  is  turned  towards 
Species  of  the  left,  or  towards  the  incident  rays,  and  a  negative  to  that  whose  convexity  is  turned  to  the  right,  or  from 
distil?  ^(sh  tnem>  we  sna"  nave  tne  f°"owinn  varieties  of  denomination  : 

braically  meniscus  ")    Tboth  radii  +,  as  fig.  46,  49,  a,  or 

concavo-convex  j    (.both  radii  — ,  as  fig.  46,  49,  6, 

pi-  n        n          _fradius  of  first  surface  +,  of  second  infinite,  fig.  45,  b, 
(.radius  of  first  surface  infinite,  of  second  — ,  fig.  45,  a, 

plano-concave  /radius  of  first  surface  — ,  of  second  oo ,  fig.  48,  />, 
(.radius  of  first  surface  cc ,  of  second  +  ,  fig.  48,  a, 

double-convex  :   radius  of  first  surface  +,  of  second  — ,  fig.  44, 
double-concave:  radius  of  first  surface  — ,of  second  +,fig.  47, 

the  rays  being  supposed  in  all  cases  to  pass  from  left  to  right. 

A  compound  lens  is  a  lens  consisting  of  several  lenses  placed  close  together. 
An  aplanalic  lens  is  one  which  refracts  all  the  rays  incident  on  it  to  one  and  the  same  focus. 
2gi  Problem.   To  find  the  power  and  foci  of  a  single  thin  leiu  in  vacua. 

Focus  of  a       Let  R-  and  R"  be  the  curvatures  of  its  first  and  second  surfaces  respectively,  ft  the  refractive  index  of  the 
single  lens.  j 

medium  of  which  it  consists,  m  =  —  ;  F  its  power :  then  we  have,  since  the  last  refraction  is  made  into  vacuum, 

F  =  /iF  +  F";        /=F  +  D; 
but,  F'=  (1   -  m')R',  and  F"  =  (1  —  m")  R" ;  and  as  fi'  = and   m"  =  /.,  these  become   respectively 

—  (ji  —  1)  R'  and  —  (fi  —  1)  R'',  so  that  the  foci  of  the  lens  are  finally  determined  by  the  equations 


Fundamen-  F  =  (,  -  1)  (R'  -  R")l 

talequations.  f  =  F  +  V  J 


talequations.  f  =  F  + 

262.  Carol.  1.  The  power  of  a  lens  is  proportional  to  the  difference  of  the  curvatures  of  the  surfaces  in  a  meniscus 

Power  of  a  or  concavo-convex  lens  ;   and  to  their  sum,  in  a  double-convex  or  double-concave. 

In  plano-convex,  or  plano-concave  lenses,  the  power  is  simply  as  the  curvature  of  the  convex  or  concave 
surface. 

•263.  Carol.  2.   In  double-convex  lenses  R'  is  positive  and  R"  negative,  so  that  when  /*  >  1,  F  is  positive,  or  the 

rays   converge  to  a  focus  behind  the  lens.     In  plano-convex,    R"  =  0  and  R'  is  +  ;  or   R'  =  0  and   R"  is 

negative,  (260)  ;    hence  in  both  cases  F  is  positive  and   the  rays  also  converge.     In  meniscus  lenses  also, 

R'is  +,  and  R", though  +,  is  less  than  R',  (fig.  49 ;)  therefore  in  these,  also,  the  same  holds  good.     In  all  these 

Real  and     cases  tne  f°™*  **  sa'd  to  be  real,  because  the  rays  actually  meet  there.     In  double-concave,  plano-concave,  or 

virtual  foci,  concavo-concave  lenses,  the  reverse  holds  good  ;    the  focus  lies  on  the  opposite  side,  or  towards  the  incident 

rays,  and  parallel  rays,  after  refraction,  diverge  from  it.     In  this  case,  therefore,  they  never  meet,  and  the  focus 

is  called  a  virtual  focus. 

264.  Coral.  3.  If  /*  be  <  1,  or  the  lens  be  formed  of  a  medium  rarer  than  the  ambient  medium  (which  need  not  be 

vacuum,  provided  the  whole  system  be  immersed  in  it,)  /»  —  1  is  negative,  and  all  the  above  cases  are  reversed. 
In  this  case  convex  lenses  give  virtual,  and  concave,  real  foci. 

265  Carol.  4.     For  lenses  of  denser  media,  the  powers  of  double-convex,  plano-convex,  and  menisci  are  positive  ; 

Positive  and  and  those  of  double  plano-concave  and  concavo-convex  lenses,  negative  ;  vice  versa  for  rarer  media. 
negative  Carol.  5.    The  focus  of  parallel  rays  is  at  the  same  distance,  on  whichever  side  of  the  lens  the  rays  fall.     For 

powers.        if  the   lens  be  turned  above,  R'  becomes  R'',  and  vice  vend;  but,  since  they  also  change  their  signs,  F  remains 

266.  unaltered. 

267.  Carol.  6.    The  equation/^:  F  +  D  gives  df  =  dD.     This  shows  that  the  foci  of  incident  and  refracted  rays 
Conjugate    move  always  in  the  same  direction,  if  the  former  be  supposed  to  shift  its  place  along  the  axis ;  and,  moreover, 
foci  move  m  that  their  proximities  to  the  lens  vary  by  equal  increments  or  decrements  for  each. 

Problem.  To  determine  the  central  foci  of  any  system  of  lenses  placed  close  together,  the  lenses  being  supposed 

infinitely  thin. 

Central  tori  The  g6"61"11'  problem  of  a  system  of  spherical  surfaces  contains  this  as  a  particular  case ;  for  we  may  regard 
of  a  system  the  posterior  surface  of  the  first  lens,  and  the  anterior  of  the  second,  as  forming  a  lens  of  vacuum  interposed 
of  thin  lenses  between  the  two  lenses,  and  so  for  the  rest.  Thus  the  system  of  lenses  is  resolved  into  a  system  of  spherical 
m  contact,  surfaces  in  contact  throughout  their  whole  extent ;  the  alternate  media  having  their  refractive  indices,  or  the 

alternate  values  of  M,  unity.     If  then  we  call  fi',  fi",  fi'",  &c.  the  refractive  indices  of  the  lenses,  we  shall  have 

M  =  l;         M '  =  /•';          M"=l;         M'"  =  ,u";         Mu  =  1,  &c. 


LIGHT. 


383 


Light     The  compound  power  F  then  will  (258,  o)  be  represented  by 

'    '  p  =/t'F'  +  F"  -rV'F'"  +  Fiv 


Fvi  +,  &c. 


But 


1 


F'  =  (1  -  TO')  R'  =  --  (/.'  -  1)  R' 
F"  =  (1  -  TO")  R"  =  (1  -  yu')  R", 


because  TO'  =  — —  and  m"  =  /*'.     Consequently, 

p1  F'  +  F"  =  (ji>  -  1)  (R'  -  R") 
and  similarly 

pit  F'"  +  F"  =  (n']  —  1)  (R'"  -  •  Riv),  &c. 
so  that  we  get,  finally, 

F  =  (/  -  1)  (R'  -  R'')  +  0«"  -  1)  (R'"  -  Riv)  +  &c. 

Now,  the  several  terms  of  which  this  consists  are  (by  Art.  261)  the  respective  powers  of  the  individual  lenses 
of  which  the  system  consists,  so  that  if  we  put  (according  to  the  same  principle  of  notation)  L',  L",  L'",  &c. 
for  the  powers  of  the  single  lenses,  and  L  for  their  joint  power  as  a  system,  we  have 


Part  F. 


Superposi- 
tion of 
powers. 
Power  of  a 
system  of 
lenses  is  the 
sura  of  the 
powers  of 
the  compo- 
nent indivi- 
duals. 


L  =  L'  +  L"  +  L"'  +,  &c. 


(-7) 


an  equation  which  shows  that  the  power  of  any  system  of  lenses  is  the  sum  of  the  powers  of  the  individual  lenses 
which  compose  it  ;  the  word  sum  being  taken  in  its  algebraic  sense,  when  any  of  the  lenses  has  a  negative  power. 
Moreover  it  is  easy  to  see  that  we  also  have/"  =  L  +  D,  as  in  the  case  of  a  single  lens. 

Reciprocally,  we  may  regard  a  system  of  spherical  surfaces  forming  the  boundaries  of  contiguous  media  (as 
in  the  instance  of  a  hollow  lens  of  glass  enclosing  water)  as  consisting  of  distinct  lenses,  by  imagining  the 
concavity  of  one  medium  and  the  convexity  of  that  in  immediate  contact  with  it  separated  by  an  infinitely  thin 
film  of  vacuum,  or  of  any  medium  having  its  surfaces  equicurve,  as  in  fig.  50  ;  and  thus  a  system  of  any  number 
(n)  of  media,  whose  surfaces  are  in  contact  throughout  their  whole  extent,  may  be  conceived  replaced  by  an  equi- 
valent system  of  2  n  —  1  lenses,  the  alternate  ones  being  vacuum,  or  void  of  power.  This  way  of  considering 
the  subject  has  often  its  use.  It,  moreover,  leads  to  the  result,  that  the  power  of  any  system  of  spherical 
surfaces  placed  in  vacua  is  the  sum  of  the  powers  of  the  several  lenses  into  which  it  can  be.  resolved,  each  placed  in 
vacuo  and  acting  alone. 

Let  us  now  return  to  the  case  of  surfaces  separated  by  finite  intervals  ;  and,  first,  let  us  inquire  the  foci  of  a 
system  of  surfaces  separated  by  intervals  so  small  that  their  squares  may  be  neglected.  In  this  case  the  equa- 
tions (j),  Art.  251,  become  simply 

D'=D;         D"  =  /'  +  /"<',-        D'"  =  /"  +  /"«*",  &c.; 


269 


and  substituting  these  values  in  the  equations  (z),  and  retaining  the  notation  of  Art.  257,  we  find 
MW«  =  M'F'  +  M"F"  +  ....  M«FM  +  D 


Power  of  a 
system  of 
spherical 
surfaces 
expressed. 
270. 
Foci  of  a 
system  of 
surfaces  se- 
parated by 
small  finite 
intervals. 


Now  in  this  we  are  to  consider  that 

/'  =  F'  +  m'D,  /"  =  F»  +  m"F'  +  m'm"D',  &c. 
and  the  values  of  /',/",  &c.  so  expressed,  being  substituted  in  the  foregoing  equation,  we  find 

M/=  M'  F'  +  M"  F"  +  M1"  F'"  +  &c. .  .  +  D  (r) 

+  M'  (F'  +  m'  D)2  t'  +  M"  (F"  +  TO"  I"  +  m"m'D)*  t"  +,  &c. 
Carol.  In  the  case  of  two  surfaces,  supposing  M  =  1,  or  in  the  case  of  a  single  lens  in  vacuo,  this  gives 

CO 


/=  0,  -1)  (R'  -  R")  +  D  +  —  {  0*  -  1)  R'  +  D  }  *  t. 
For  parallel  rays,  this  becomes 


271. 
Case  of  a 
single  lens 
of  small  but 
finite  thick- 


F  =  fc  -  1)  (R'  -  R")  + 


R'*.  t; 


(0 


t  being  here  put  for  t',  the  interval  between  the  surfaces  or  total  thickness  of  the  lens. 

Problem.  To  determine  the  foci  of  a  lens,  whose  thickness  t  is  too  considerable  to  allow  of  any  of  its  powers 
being  neglected. 

Here  we  must  take  the  strict  formulae 


D'  =  D  ;         D"  = 


/'=(!-  TO')  R'  +  TO'  D ;         and  /*='!-  m")  R"  +  m"  D" 


272. 

Focxis  of  ?. 
lens  of  any 

thickness. 


The  latter  equation  gives,  on  substitution,  and  recollecting  that  mf  r=  —  =  m  and  m"  =  ftt 


384  LIGHT. 


/  =  /"=-  -j-  -  i  («) 

1  -  —  {  (/<  -  1)  R'  +  D  }  * 

and  for  parallel  rays 

,,  0,  -  1)  (R'  -  R")  +  0,  -  !)•  .  R'  R"  t 
p-i.(p-l)R' 

273.  Example  1  .   To  determine  the  foci  of  a  sphere. 
Foci  of  a  _ 

sphere  Here  R/,  _  _  R,  _  _  R  t  =  —  ;         and  the  equations  (u)  and  (v)  become 

R 

(2A.-8)R+(2-AQD  2^-2 

(2-X)R-2D~  ^^T'R 

274.  Core/.  1.  If  /«  =  2,  for  instance,  these  values  become 


In  this  case,  then,  since  /and  F  express  the  proximities  of  the  foci  to  the  posterior  surface  of  the  sphere,  we 
see  that  the  focus  for  parallel  rays  falls  on  this  surface,  and  that  in  any  other  case  (as  in  fig.  51  and  52)  q  is  given 

by  the  proportion  QC:CE::EH:H<7. 

275.  Carol.  2.  Whatever  be  the  value  of  ft,  the  focus  for  parallel  rays  after  the  second  refraction  bisects  the  distance 
between  the  posterior  surface  of  the  sphere,  and  the  focus  after  the  first  refraction. 

276.  Example  2.  To  determine  the  foci  of  a  hemisphere,  in  the  two  cases  ;  first,  when  the  convex,  secondly,  when 
Foci  of  a     the  plane  surface  receives  the  incident  light. 

hemisphere.  i 

In  the  first  case,  R'  =  R  ;          R"  =  0  ;  t  =  —  :  therefore  we  find 

Qx-DR  +  D 

R-D  =  0*- 

277  In  the  other  case,  when  the  rays  fall  first  on  the  plane  side,  R'  =  0,  R"  =  —  R,  and  t  =  —  ,  so  that 


If  the  thickness  of  a  spherical  segment  exposed  with  its  convex  side  to  the  incident  rays  be  to  the  radius  as 
*7o 

u  to  u  —  1,  or  if  t  =      **  , .  -=r-  =  7-, m,  an(*  R"  =  0.  the  expressions  (u)  and  (v)  become 

p,  —  1     R        (1  —  m)  K 

D 

In  this  case  the  focus  for  parallel  rays  falls  on  the  posterior  surface  of  the  segment. 
279.          In  general,  for  any  spherical  segment,  if  exposed  with  its  convex  side  to  the  rays,  R"  =  0,  and 


nsp-  ,_  Q.-DR  +  D  p=          A.  fr  -  1)  R 

cal  segment,  J~P"a+{(u-l)R+D}t'  /«  +  (/*—  1)  R  t 

convex  side 

first-  If  the  plane  side  be  exposed  to  the  rays 

Plane  side 


F=0*-1)R. 


280.  If  R1  =  R     o;  if  the  lens  be  a  spherical  lamina  of  equal  curvatures,  the  one  convex,  the  other  concave, 

Focus  of  a 
spherical  /t  D  +  Qt  -  1)  {  (/t  -  1)  R  +  D  }  R  <  (/a  -   1)*  R'  t 

oflqual  ,.-{f>-l)R+DM  '    M  -  0»  -  1)  B<  ' 

curvatures. 


LIGHT. 

Light. 
—  v—  • 

§  X.    Of  the  Aberration  of  a  System  of  Spherical  Surfaces. 

Problem.  To  determine  the  focus  of  any  annulus  of  a  spherical  refracting  or  reflecting  surface.  281. 

The  equations  (a)  of  Art.  244,  of  the  last  section,  in  fact,  contain  a  general  solution  of  this  problem  ;  bat  Focus  of  a 
the  applications  of  practical  Optics  require  an  approximate  solution  for  annuli  of  small  diameter,  or  in  which  y  sma"  an»u- 
is  small  compared  with  r.     Conceiving  y,  then,  so  small  that  its  fourth  and  higher  powers  may  be  neglected,  the  ^erLal 
expressions  in  the  article  cited  give  surface  in- 

/-;  -  T  3/3  y*  vestlgated. 

x  =  a  —  vr-      y*  —  a  —  r+-^—;  a  —  x  =  r  --  i  — 


yZ=,r(a       r)  + 

and  substituting  these  in  the  value  of  C  q,  found  in  the  same  article,  we  get  for  the  distance  of  the  focus  of 
refracted  rays  from  the  vertex 

—  _      ftr(r-  a)  _  ft  -  1  __  ga  (a  +  ft  r)  __  y*_ 

a  —  /»  a  +  /•  r  2/4      '    (a  —  r)  (a  —  ft  a  +  p  r)s    '     r 

In  conformity,  however,  with  the  system  of  notation  adopted  in  the  last  section,  instead  of  expressing  directly      g82 
C  q,  we  will  take  its  reciprocal.     As  we  have  hitherto  represented  the  value  of  this  reciprocal  for  central  rays 
by  f,  we  will  continue  to  do  so  ;  and  for  rays  incident  at  the  distance  y  from  the  vertex,  we  will  represent  the 
same  reciprocal  by  /+  A  /;   A  f  then  will  be  vhat  part  of  /due  to  the  deviation  of  the  point  of  incidence  from 
the  vertex.     Now,  neglecting  y4,  we  have 

1       _    a  —  pa  +  pr         />  -  1       a3  (a  +  ftr) 

'         O    ..  3         '         ~  3    /„  _\  .1  a  *    ' 


Cq             pr(r  —  a)             2  p*          r3  (a  -  r)3 
Now    if  we   put,    as   we    have    hitherto   done,    /a,  =  ,  r  =  — ,  a  =  — — ,   and  substitute  these 

Tfti  Jv  IV  \J 

in  the  above,  we  shall  get  the  value  of  — — ,  or  off  +  A  f,  in  terms  of  m,  R,  and  D ;  and  from  this,  subtracting 

Cg 

the  term  independent  of  y2,  which  is  the  value  of  f,  we  shall  get  A  /as  follows, 

A/=    m(l~m)  (R-D)«{«R-0  +  ro)D}y«.  (c) 

m 

Definition.  The  longitudinal  aberration,  is  the  distance  between  the  focus  for  central  rays  and  the  focus  q  of  283. 
the  annulus,  whose  semidiameter,  or  aperture,  is  y  =  M  P.  Longitudi- 

The  lateral  aberration  at  the  focus,  is  the  deviation  from  the  axis  of  the  refracted  ray,  or  the  portion  fk,  na'  and 
intercepted  by  the  extreme  ray,  of  a  perpendicular  to  the  axis  drawn  through  the  central  focus. 

Carol.  These  aberrations  are  readily  found  from  the  value  of  A  f  above  given ;    for  since  C  q  •=  —y,  we  <*ef'ned- 

/  284. 

1                    A/  Relation 

have  A  C  q  (=  longitudinal  aberration)  =  A  —^  •=. ^~  5  or»  calling  u>  this  aberration,  between 

/                     J  them  and 


=          A/  . 


and  since  C  q  :  qk  : :  y  :fk,  or  —  :  to  :  :  y  :  fk, 

we  have  fk,  or  the 

lateral  aberration  =  /.  y  .  u>  =  —        J    .  y ;  (e) 

where  /  =  (1  —  m)  R  +  m  D. 

Thus  the  whole  theory  of  aberration  is  made  to  depend  on  the  value  of  A  /»  an^  we  come  therefore  to  con- 
sider the  various  cases  of  this  which  present  themselves. 

Case  1.  For  parallel  rays  D  =  0 ;  and,  therefore,  285 


lateral  aberration  ss  —         •    R*  y* 

t.  IV.  S  B 


386 


LIGHT. 


Light.          Cote  2.  In  reflectors,  m  =  ft  =  —  1,  and 


Parti. 


286. 
Case  of 
reflectors. 


R  (R-  D)s 


(2R-D)* 
lateral  aberration  =  —  £  (R  —  D)2  ys. 


Or) 


which,  for  parallel  rays,  become 


287. 

Aplanatic 
foci  defined 
and  inves- 
tigated. 


In  the  general  case,  if  we  put  either  D  =  R,  or 


lateral  aberration  =  —  i  R* 


(A) 


7TL 

m  R  —  (1  +  m)  D  =  o,  which  gives  D  = R ; 


1 
R 


288. 
Aberration 
shortens  the 
focus  for 
parallel  rays 

289. 

Effect  of 
aberration 
m  other 
cases. 
Fig.  54. 
290. 


291. 

Aoerration 
of  any 
system  of 
spherical 
surfaces  in 
contact. 


the  value  of  ^  /  and  therefore  of  the  aberration,  vanishes.  The  former  case  is  that  of  rays  converging  to 
the  centre  of  curvature,  in  which,  of  course,  they  undergo  no  refraction.  In  the  latter,  the  point  is  the 
same  with  that  already  determined,  Art.  234.  It  is  evident,  from  what  was  there  demonstrated,  that  every 
spherical  surface,  C  P,  has  two  points  Q,  q  in  its  axis,  so  related,  that  all  rays  converging  to  or  diverging  from 
one  of  them,  shall  after  refraction  rigorously  converge  to  or  diverge  from  the  other.  These  points  maybe  called 
the  aplanatic  foci  of  the  surface  ;  and,  to  distinguish  them,  Q  may  be  called  the  aplanatic  focus  for  incident,  and 
q  for  refracted  rays.  To  find  them  in  any  proposed  case,  in  the  axis  of  any  proposed  surface  C,  and  on  the 

concave  side  of  the  surface,  take  C  Q  =  (/t  +  1)  X  radius  C  E  of  the  surface,  and  Cq  =  ( ^  0  x  rac"us- 

Then  will  Q  and  q  be  the  aplanatic  foci  required.  In  the  case  of  reflexion,  when  /"  =  —  l,CQ  =  Cg=0,  and 
both  the  aplanatic  foci  coincide  with  the  vertex  of  the  reflector. 

Let  us  next  trace  the  effect  of  aberration  in  lengthening  or  shortening  the  focus,  for  all  the  varieties  of  position 
of  the  focus  of  incident  rays ;  and,  first,  when  D  =  0,  or  for  parallel  rays,  A  f  is  of  the  same,  and  therefore  u> 
of  the  contrary  sign  with  R,  and  therefore  with  F,  which  is  equal  to  (1  —  m)  .  R.  Hence  it  is  evident,  that  the 
effect  of  aberration  in  this  case  must  be  to  shorten  the  focus  of  exterior  rays. 

Q  in  this  case  is  infinitely  distant.  As  it  approaches  the  surface,  or  as  the  rays  from  being  parallel  become 
more  and  more  convergent,  or  divergent,  the  aberration  diminishes ;  but  the  focus  of  exterior  rays  is  still  always 
nearer  the  surface  than  that  of  central,  till  Q  comes  up  to  the  aplanatic  focus  PL  for  incident  rays  on  the  concave, 
or  to  the  focus  F  of  parallel  rays  on  the  convex  side.  When  Q  is  at  the  former  of  these  points,  the  aberration  is 
0  ;  at  the  latter,  infinite. 

When  Q  is  situated  anywhere  between  these  points,  however,  the  reverse  is  the  case,  and  the  effect  of  aberra- 
tion is  to  throw  the  focus  for  exterior  rays  farther  from  the  surface  than  that  for  central  ones.  These  results  are 
easily  deduced  from  the  consideration  of  all  the  particular  cases,  and  hold  good  for  all  varieties  of  curvature,  and 
for  refracting  media  of  all  kinds.  In  reflectors,  the  aplanatic  foci  coincide  with  the  vertex.  In  these,  the  focus 
for  exterior  rays  is  shorter  than  for  interior  in  every  case,  except  when  the  radiant  point  is  situated  between  the 
surface  and  the  principal  focus  on  the  concave  side  of  the  reflecting  surface  ;  but  between  these  points,  longer. 

Problem.    To  determine  the  aberrations  of  any  system  of  spherical  refracting  surfaces  placed  close  together. 

Retaining  the  notation  of  Art.  257,  let  us  suppose  the  ray,  after  passing  through  the  first  surface,  to  be  incident 
on  the  second.  Its  aberration  at  this  will  arise  from  two  distinct  causes :  first,  that  after  traversing  the  first 
surface,  instead  of  converging  to  or  diverging  from  the  focus  for  central  rays,  its  direction  was  really  to  or  from 
a  point  in  the  axis  distant  from  that  focus  by  the  total  aberration  of  the  first  surface  ;  and,  secondly,  that  being 
incident  at  a  distance  from  the  vertex  of  the  second  surface,  a  new  aberration  will  be  produced  here,  which  (being, 
as  well  as  the  other,  of  small  amount)  the  principles  of  the  differential  calculus  allow  us  to  regard  as  independent 
of  it,  and  which  being  computed  separately,  and  added  to  it,  gives  the  whole  aberration  of  the  two  surfaces 
regarded  as  a  system.  The  same  is  true  of  the  small  alterations  in  the  values  of  /',  f",  &c.  produced  by  the 
aberrations.  If  then  we  denote  by  S  f"  the  change  in  the  value  of  f",  produced  by  the  action  of  the  first 
surface,  and  by  S'f",  that  arising  immediately  from  the  action  of  the  second,  and  by  A  f",  the  total  alteration 
produced  by  both  causes,  we  shall  have 

A /"=«/"+  &'f" 

Now,  first,  to  investigate  the  partial  alteration  Sf"  arising  from  the  total  alteration  A/'  in  the  value  of/',  or 
from  the  aberration  of  the  first  surface,  we  have 


since,  in  this  case, 


/"  =  (1  —  m)  .  R"  +  m"/'.  and  therefore  Sf"  =  m"  A/', 
D'=D,     D"  =  /',     D"'  =  /",  &c. 


Again,  to  discover  the  partial  variation  S'f"  in  f",  arising  immediately  from  the  action  of  the  second  surface, 
we  have,  by  the  equation  (c)  at  once,  putting/'  for  D",  and  neglecting  y"1,  &c. 


(R"  - 


m"  R"  - 


• 
but  we  have,  by  the  same  equation,  also 

If"  =  m"  A  //  _    m"  m'  &  ~  m>)    (R'  _  D).  {  ,„'  R'  -  (l  +  TO')  D  }  y\ 


LIGHT. 


387 


Light.      Consequently,  uniting  the  two,  we  have  the  value  of  A  /"•     Similarly,  the  value  of  A  /'"  may  be  derived  from     Part  I. 
— v — '  that  of  A  /",  by  a  process  exactly  the  same,  and  which  gives  — ~v~* 

anllt  /I  m  m\ 

A/"'  =  m'"  A/"  H — — (R'"  —/")*{«»'" R"'—  (1  +  TO'")/"  }y4, 

and  so  on.  Calling,  then,  as  in  Art.  257,  M',  M",  M"'. .  . .  M<">  the  absolute  refracting  indices  of  the  several  media 
into  which  the  successive  refractions  are  made,  and  putting  M (n)  =  M,  we  shall  have  no  difficulty  in  arriving  at 
the  following  general  expression,  where  A  /  denotes  the  total  effect  of  aberration  on  the  value  of  f,  the  reciprocal  General 


focal  distance  of  the  system, 


expression 
for  A/. 


M.  A/=<( 


(R'  -  D)*  {  m1  R'  -  (1  +  m')  D  } 


+  M"  .  m//(1o    m<>)  (R"  -/")«{  m"  R"  -  (1  +  m11)  f  } 


M'". 
&c. 


OT 


(0 


Successive 
values  of  f. 


(3) 


in  which  it  will  be  recollected  that 

/'  =  (!-  m')    R1    +  m1  D 

/"  =  (1  -  m")  R"  +  m"  (1  -  m')  R'  +  m'  m"  D 

/'"  =  (1  -  m'")  R'"  +  m"1  (1  -  m")  R"  +  m"'  m"  (1  -  m')  R'  +  m'"  m"  m'  D  (  ' 

&c. 

and  these  values  being  substituted  give,  if  required,  an  explicit  resulting  value  of  A  f  in  terms  of  the  radii  and 
refractive  indices,  or  their  reciprocals,  of  the  surfaces. 

If  the  system  of  surfaces  be  placed  in  vacuo,  or  the   last  refraction  be  made  into  vacuum,  M  =  1,  and  the      292. 
second  member  of  the  equation  (f)  exhibits  simply  the  value  of  A  f.     In  all  cases,  the  aberration  u>  is  given  as 
before  by  the  equation 

to  =  —  — TJ~>  and  the  lateral  aberration  is -4—  y. 

J  J 

To  express  the  aberration  of  any  infinitely  thin  lens  in  vacuo,  let  the  terms  of  the  general  equation  be  denoted       293. 
respectively  by  Q',  Q",  &c.,  so  as  to  make 


M  .  A  /  =  {  Q'  +  Q"  +  Q'"  +,  &c.  }  y«. 

le  lens   in   vacuo,   when  m"  =  —  -,  M'  = 

m 

A  /=  Q'  +  Q"  ;   and  putting,  for  a  moment,  R'  —  D  =  B,  R'  —  R"  =  C,  we  find 


(k) 

. 

Then,   for   the   case   of  a   single  lens   in   vacuo,   when  m"  =  —  -,  M'  =  —  7,  M"  ;=  1,  M  =  1,  we  have 

m  m! 


Aberration 
t^f* 

thin  lens. 


Q"=- 


1  -  m' 

2m'3  • 


whence 


Q'  +  Q"  = 


-    — 


/*  (m'  B  —  C)  *  {  m'2  B  -  m'  D  —  C  } 
C  {  (2  m'  B  -  C)  (m'2  B  -  m'  D)  +  (C  -  m'  B)*  } 


The  expression  in  brackets,  putting  for  B  and  C  their  values,  and  —  for  m1,  will  become 

-JT  {  ((2  ~  A  R'  +  /.  R"  -  2  D)  (R'-  (1  +  ^)  D)  +  AI  (0.-1)  R'-^R"  +  D)«}. 
If  now  we  multiply  out,  arranging  according  to  powers  of  D,  and  substitute  the  result,  as  also  the  value  of  mf, 
(=  —  ,)  and  of  C,  (  =  R'  -  R",)  in  Q'  +  Q",  or  A  /,-  we  get 


A  /=  Ou-  1)  (R1- 


where 


a  =  (3  —  2  n*  +  p3)  R'*  +  (ft  +  2 

0  =  (4  +  3  A  -  3  /.*)  R'  +  0*  +  3  /*")  R" 

<v  =  2  -t-  3  ft 


{«  -/3D  +  7D3} 

-  2  Hs)  R'  R"  +  /»'  R"2< 


General 
expressio* 
for  it 


(0 


3  E  2 


388  LIGHT. 

Light.     Now  it  has  been  shown,  (Art.  261,)  that  (/*  —  1)  (R1  —  R")  expresses  the  power  of  the  lens,  so  that,  putting  L      Part  I. 
—*s~~S  for  this,  we  have 


7D^)yi.  (m) 

Such  then  is  the  general  expression  for  A  /,  the  fundamental  quantity,  from  which  the  aberration  u>  may  be  had 

A  f 
in  any  lens  by  the  equation  ia  =  --  ~ft[~- 

294  Carol.  I.  The  aberration  of  a  lens  vanishes  when  D  is  so  related  to  R1,  R"  and  /i,  as  to  give 

Cases  in  a  _i_  */~al  -  ~A  - 


of  a  single 

lens  can  be  Now  we  find,  by  substitution  and  reduction, 

vanish!"  /3»  -  4  a  7  =  ^  {  (R'  +  R  •')•  -  (8  ,»  +  3  /.»)  (R1  -  R")  «  } 

and  unless  this  quantity  be  positive,  that  is,  unless 


the  focus  of  incident  rays  cannot  be  so  situated  as  to  render  the  aberration  nothing.  But,  if  the  curvatures  R' 
and  R"  of  the  surfaces  be  such  as  to  satisfy  this  condition,  the  value  of  D  may  be  calculated  at  once  from  the 
equation  (k.) 

295.  Carol.  2.  Whenever,  in  meniscus  or  concavo-convex  lenses,  the  difference  of  the  curvatures  of  the  surfaces  is 
small  in  comparison  with  their  sum,  that   is,  whenever  a  moderate  focal  length  is  produced  by  great  curvatures, 
the  aberration  admits  of  being  rendered  evanescent  by  properly  placing  the  focus  of  incident  rays.     In  a  lens  of 
crown  glass  where  /n  =  1.52,  we  have  ^2  ft  +  3  p*  =  3.16 ;   therefore  the  sum  of  the  curvatures  must  be  at  least 
3.16  times  their  difference,  to  satisfy  the  condition  of  possibility.     In  double-convex  or  double-concave  lenses,  R1 
and  R'1  having  opposite  signs,  the  condition  can  never  be  satisfied. 

296.  Carol.  3.   If  a  =  0,  the  aberration  vanishes  for  parallel  rays.     This  condition  is,  however,  only  to  be  satisfied 
No  known    by  real  values  of  R'  and  R''  when  ft,  is  equal  to  or  less  than  j,  and  no  such  media  are  known  to  exist. 

medium  can  Carol.  4.  The  effect  of  aberration  will  be  to  shorten  or  lengthen  the  focus  for  exterior  rays,  according  as  the 
render  the  sign  of  A  /is  the  same  as,  or  the  opposite  to,  that  of  f.  In  particular  cases  it  will,  of  course,  however,  depend 
nothintl0for  on  t'le  vames  °?  /*>  R»  &'«  and  D  which  shall  take  place.  The  principal  case  is  that  of  parallel  rays,  in  which 
paralldraysD  =  °>  and 

CJjJli;  A/=  -f-  •  L  {  (2  -  2  p*  +  p°)  R"  +  0*  +  «,*•  -  «  /•')  R'R"  +  P*  R"*  } 

which  the 

aberration  an(j  fae  focus  of  external  rays  will  be  shorter  or  longer  than  that  of  central  ones,  according  as  this  quantity  has 

iengrtiens"  *e  *awle>  or  opposite  sign  with  L,  that  is,  according  as 

the  focus-  (2  -  2  /.«  +  /*3)  R"  +  G»  +  2  /»*  -  2  /t3)  R1  R"  +  /»'  R''s 

is  positive  or  negative.  Now,  from  what  we  have  already  seen  in  the  last  corollary,  this  quantity  never  can  be 
rendered  negative  by  any  real  values  of  R'  and  R",  unless  /u  be  less  than  J.  For  all  other  media,  therefore, 
(comprehending  all  yet  known  to  exist  in  nature,)  every  lens,  whatever  be  the  curvatures  of  its  surfaces,  has  the 
exterior  focal  length  for  parallel  rays  shorter  than  the  central. 

298.  Carol.  5.    In  a  glass  meniscus,  when  the  radiant  point  is  on  the  convex  side,  and  the  rays  diverge,  we  have 
Qase  of  a     4+3/t  —  3/»*a  positive  quantity ;  and,  R'  and  R"  being  both  positive,  /3  is  so ;  hence  (D  being  negative  in 
glass            this  case)  the  term  —  /3  D,  and  therefore  the  whole  factor  a  —  /JD  +"/D*  is  positive;    and  L  being  also 
meniscus,     positive,  A  f  is  so ;    and,  therefore,  w,  the  aberration,  negative.      Hence,  when  Q   is   beyond   F,  the  focus 

for  parallel  rays  incident  the  other  way,  the  exterior-focus  is  the  shorter ;  but  when  between  F  and  C,  the 
longer. 

/  R '    I    R ''   \  ^ 

299.  Carol.  6.    Unless    (  — ; ^77-)     >  2  /t  +  3  u2,  no  real  value  of  D  can  render  a  — /3D  +  7  D2  negative. 

Rule,  for  a  V  R   -  R  '   / 

oVusnses'to      **  aPPears>  therefore,  that  in  all  double-convex  or  concave  lenses,  as  well  as  in  all  meniscus  and  concavo-convex 

effect"of      ones»  *n  which  the  sum  of  the  curvatures  of  the  surfaces  is  greater  than  */  2  ft  +  3  /**  times  their  difference,  the 

aberration     factor  a  —  ft  D  +  ff  D*  is  positive  for  all  values  of  D,  and  therefore  the  aberration  <o  has  in  all  such  lenses  the 

in  lengthen-  s'gn  opposite  to  that  of  L.     Hence,  for  all  such  lenses,  we  have  the  following  simple  and  general  rule :  the  effect 

ing  or  short-  of  aberration  will  be  to  throw  the  focus  of  exterior  rays  more  TOWARDS  the  incident  light  than  that  of  central 

S  tn«     ones,  when  the  lens  is  of  a  positive  character,  or  makes  parallel  rays  CONVERGE,  but  more  FROM  the  incident  light 

if  of  a  negative,  or  if  it  cause  parallel  rays  to  DIVERGE. 

300.  Coral.  7.  All  other  lenses  have,  as  in  the  case  of  single  surfaces,  aplanatic  foci,  corresponding  to  the  roots  of 
the  equation  a—  £D  +  7D4=0.     In  general  there  are  two  such  foci  of  incident  and  two  of  refracted  rays :  and 


LIGHT.  389 

Liglit.      rules  might  easily  be  laid  down  for  determining  in  what  positions  of  the  radiant  point,  with  respect  to  these  foci      Part  I. 
— .,^~ —  and  the  lens,  the  aberration  tends  to   shorten   or  lengthen  the  exterior  focus ;  but  it  is  simpler  and  readier  to  •  — y  —^  > 
have  recourse  at  once  to  the  algebraic  expressions.  How  to 

Carol.  8.  In  the  case  of  reflexion,  as  when  rays  are  reflected  between  the  surfaces  of  thin  lenses  of  transparent  Procce(J  1D 
media,  we  have  m  =  m"  =  &c.  =  /  =  /*"  =  &c.  =  —  1 ;  M1  =  —  1,  M'1  =  +  I,  &c.,  and  M  =  +  1,  accord-  oih^™ei- 
ing  as  the  number  of  reflexions  is  even  or  odd ;  therefore  for  n  reflexions  we  have  Case  Of'le. 

flexion  be- 

/'    =  2  R'     —  D  ~)  tween  any 

/•n_onii          on'-LD  /  system  of 

/      —  *  U  \  .  (-)  transparent 

f"  =  2  R'"  —  2  R"  -(-   2  R'  —  D    f  '  surfaces. 

&c.  ) 

and 

I  -R"  (R"  -2R'  +  D)» 

A  f—  ( n"+'< 

J  ~  j  +  R"1  (Rm  -  2  R"  +  2  R'  -  D)* 

(.-&c. 

which  formulae  serve  to  determine,  in  all  cases  of  internal  reflexion  between  spherical  surfaces,  both  the  places 
of  the  successive  foci  and  the  aberrations. 

Corol.  9.    If  the  reflexions  take  place  between  equicurve  surfaces,  having  their  concavities  turned  opposite      302. 
ways,/',  /",  &c.  are  in  arithmetical,  and   therefore  their   reciprocals,  or  the  focal  distances,  in  harmonic  pro- 
gression. 

Problem.  To  construct  an  aplanatic  lens,  or  one  which  shall  refract  all  rays,  for  a  given  refractive  index,  and      303. 
converging  to  or  diverging  from  any  one  given  point,  to  or  from  any  other.  General 

Let  Q  and  q  be  the  points,  the  former  being  the  focus  of  incident,  the  latter  of  refracted  rays.     Let  /t  =  index  construction 
of  refraction  ;  and  putting  Q  q  =  2  r,  and  assuming  b  any  arbitrary  quantity,  construct  the  curve  whose  equation  of  *n  ,atlla" 
is  (n),  Art.  232.     Let  H  P  C,  (fig.  36,)  be  this  curve ;  and  with  centre  q,  and  any  radius  q  N  less  than  q  P,  any  j-; '°  gg"8' 
one  of  the  refracted  rays  describe  the  circle  H  N  K.     Then  since  the  ray  Q  P,  by  the  nature  of  the  curve  H  P  C, 
is  after  refraction  directed  to  or  from  q,  and,  being  incident  perpendicularly   on   the  second  surface,   suffers 
there  no  flexure,  it  will,  if  supposed  to  emerge  from  the  medium,  here  continue  its  course  to  or  from  q.     If  then 
we  suppose  the  figure  C  P  H  N  K  to  revolve  round  Q  q,  it  will  generate  a  solid,  which,  being  composed  of  the 
proposed  medium,  is  the  lens  required.     If  the  rays  be  parallel,  as  in  fig.  38,  the  curve  H  P  C,  as  we  have  seen,  is  Fig.  38. 
a  conic  section,  which,  if  the  lens  be  denser  than  the  ambient  medium,  is  an  ellipse.     Thus,  a  glass  meniscus  lens, 
whose  anterior  convex  surface  is  elliptic,  and  posterior  spherical,  having  its  centre  in  the  focus  of  rays  refracted 
by  the  first  surface,  is  aplanatic. 

But,  without  having  recourse  to  the  conic  sections,  the  same  thing  may,  in  certain  cases,  be  accomplished  by      304. 
spherical   surfaces  only.     For  if  Q  and  q  (fig.  53)  be  the  aplanatic  foci  of  the  spherical   refracting  surface,  Case  when 
and  if  with  the  centre  q  and  any  radius  greater  than  9  C,  when  the  incident  rays  diverge  from  Q,  as  in  the  lower  of.^"!^8 
portion  of  the  figure,  but  less  if  they  converge  to  Q  as  in  the  upper,  we  describe  a  circle  K  L,  or  A;  I,  and  turn  the  natic  lens 
whole  figure  about  Q  q  as  an  axis,  the  surfaces  C  P  K  L,  or  cp  k  I,  will  generate  the  aplanatic  lens  in  question,  are  all 
This  also  follows  evidently  from  the  general  formula,  (t,  Art.  291,)  for  if  R"  =/',  the  expression  of  A  /for  the  spherical. 
lens  becomes  simply  F'S-  53' 

Iann  ' 
""~    "*  I  C  I          y-      1 

i  i  ( \\   —  \j  \  *   <  ffi  Jtv  •••  f  1  ~r  TTL  j  1 J  f  y  » 

2 

which  vanishes  when  D  = r  R',  or  when  Q  is  the  aplanatic  focus  of  incident  rays  for  the  first  surface. 

1  +  m' 

More  generally,  however,  the  equation  a  —  /3D+7D2  =  0,  assigns  the  universal  relation  between  /t,  D,  R', 
R",  which  constitutes  the  lens  aplanatic.     See  Cor.  1,  Art.  294. 

Problem.  To  assign  the  most  advantageous  form  for  a  single  lens,  or  that  which,  with  a  given  power,  has  the.       305. 
least  possible  aberration  for  parallel  rays.  Most  ad- 

Since  the  aberration  cannot  be  rigorously  made  to  vanish  for  parallel  rays,  when  u  >  ±  (Art.  296)  we  have  to  vantageous 

form  for  a 

A   f                   A.  f  single  lens 

make  it  a  minimum.     Now  w  = -~-  = — —  for  parallel  rays,  or  f°r  parallel 

f*                     L2  raysdeter- 

„                                                                             2  mined. 

<o  =s j  -  .    —  ;  and,  in  general,  d  ui  =  —  {  L  d  a  —  a  d  L  } 

In  the  present  case  L  is  given,  therefore  we  must  put  d  n  =  0,  which  gives 

0=2  (2-  2p*  +  p*)  R'dR'+  (/.+  2/.2-2/.3)  (R'  d  R"  +  R" d  R')  +  2  /•'  R"<£R". 
But  the  condition  d  L  =  0  gives  d  R1  =  d  R" ;  so  that  our  equation  becomes,  on  substitution  and  reduction, 


390  L  I  G  II  T. 

Light,      that  is  to  say 
~v~"'  R"        2^-- 


R1        2  p*  +  ft 

In  the  case  of  a  glass  lens,  taking  fi  =  1.5,  this  fraction  becomes  equal  to  —  —  ,  which  shows  that  the  lens 

must  be  double-convex,  having  the  curvature  of  the  posterior  surface  only  -  -  that  of  the  anterior,  or  its  radius 

six  times  as  great.     Artists  sometimes  call  such  a  lens  a  "  crossed  lens." 

Carol.  1.     If  ft  =  1.6S61,  as  is  nearly  the  case  with  several  of  the  precious  stones  and  the  more  refractive 
w  -Slasses>  R'/=°;   and  the  most  advantageous  figure  for  collecting  all  the  light  in  one  place  is  plano-convex, 

s    having  its  convex  side  turned  to  the  incident  rays. 

piano-  1|S 

convex  Carol.  2.  Calling  the  aberration  of  a  lens  of  the  best  figure  to,  we  shall  have  to  =  --  -  w2  .  L   for  glass 

307. 

Aberrations  whose  refractive  index  is  1.5,  and  the  proportional  aberrations  of  other  forms  will  be  as  follows: 
of  various 

species  of  Plano-convex,  plane  side  first  (or  towards  the  light)    ....    4.2  x  <« 

lenses  deter- 
mined for  Plano-convex,  curved  surface  first  ..................    1.081   x  <a 

parallel  rays  Double  equi-convex,  or  concave    ..................    1.567  x  <» 

Problem,  To  investigate  a  general  expression  for  the  aberration  of  any  system  of  infinitely  thin  lenses  placed 
oVlTs""    close  together  in  vacua. 
of  fenses.6111       The  general  expression  for  MA/,  or,  since  M  =  1  in  the  case  before  us,  of  A  /,  is 


(Q'  +  Q"  +  Q'"+  Qiv  +  ,  &c.)  y2, 

which  divides  itself  into  terms  originating  with  the  successive  lenses  in  the  following  manner, 

A  /=  (Q'  +  Q")  y2  +  (Q"'  +  Q*)  y*  +,  &c. 

The  first  of  these  quantities  we  have  already  considered  ;  let  us  now,  therefore,  examine  the  constitution  of  the 
rest.  Let  then  /*'  be  the  refractive  index  of  the  first  lens,  //'  of  the  second,  pf"  of  the  third  ;  and  let  a',  ft',  r/' 
represent  the  values  of  a,  /3,  7  for  the  first  lens,  or  the  expressions  in  (I,  292,)  writing  only  ft'  for  ft  ;  also  let 
0  I  P"i  7"  represent  their  values  for  the  second  lens,  or  what  the  same  expressions  become  when  ft"  is  put  for  ft, 
and  R'"  and  Riv  respectively  for  R'  and  R",  and  so  on  for  the  rest  of  the  lenses. 

Now  if  we  consider  the  values  of  Q'"  and  Qiv,  it  will  be  seen  that  they  are  composed  of  the  quantities  m'", 
miv,  M'",  Miv,  R'",  Riv,/"  and/'",  precisely  in  the  same  manner  that  Q'  and  Q"  are  of  m1,  m",  M1,  M",  R',  R", 
D  and/1. 

Moreover,  since  by  Art.  251  we  have 

/'  =  (1  -m')  R1  +  m'D 
/"  =  (1  -  m")  R"  +  m"  /' 

=  (1  -  TO")  R"  4-  m"  (!-*»')  R'  +  m"m'D 

=  (/•  —  1)  (R1  -  R")  +  D,  since  m'  =  -  ,  m"  =  ft. 

=  L  +  D  ;  call  this  D"  ;  (L  is  the  power  of  the  first  lens) 
/"'  =  (1  -  m'")  R'"  +  m!"  D" 
/"  =  (1  —  m'")  Riv  +  m'"f"  =  L"  +  D"  as  before  ;  (L"  is  the  power  of  the  second  lens) 

=  L  +  L'  4-  D  ;  and  so  on. 

And  it  is  clear  that  Q'"  +  Q'T  will  be  the  same  function  of,  i.  e.  similarly  composed  of,  the  refractive  index  and 
curvatures  of  the  surfaces  of  the  second  lens,  and  of  the  quantities  D"  and  /"',  that  Q'  +  Q"  is  of  the  re- 
fractive index  and  curvatures  of  the  first  lens,  and  of  D  and  /'.  It  follows,  therefore,  that  the  very  same 
system  of  reductions  which  led  to  the  equation 

Q'  +  Q"  =  ~  (a  -  ft  D  +  7  D2) 

being  pursued  in  the  case  of  Q'"  +  Qiv.  must  lead  to  the  precisely  similar  equation 

Q"1  +  Qiv  =  ^4r  («"  -  /3"  D"  +  7"  D"s) 

2/i 

General        an<^  so  on  ^or  ^e  remaining  lenses  ;  so  that  we  shall  have,  ultimately,  for  the  whole  system  (writing  L',  D',  ft 
expression     for  L,  D,  ft) 

1  +&c. 


in  which  there  are  as  many  terms  as  lenses. 


LIGHT. 

Coro/.  For  parallel  rays,          D'  =  0;         D"  =  L';         D"'  =  L'  +  L",  &c. 
/  therefore 


/*"'    a  ** 

+  &c. 

Although  the  aberration  of  a  single  lens  for  parallel  rays  admits  of  being  destroyed  only  on  a  certain  hypo-  311. 
thesis  of  the  refractive  index,  which  has  no  place  in  nature,  yet,  by  combining  two  or  more  lenses,  it  may  be 
destroyed  in  a  variety  of  ways.  Thus,  in  the  case  of  two  lenses,  the  expression  (f)  being  put  equal  to  zero, 
gives  an  equation  involving  /*',  /t",  L',  L",  R',  R",  R'",  Riv ;  or  (since  L'  and  L"  are  given  in  terms  of  /»,  p' 
and  R',  R",  &c.  and  since  /»',  /*"  are  given  quantities)  only  the  four  unknown  quantities  R1,  R",  R'",  R''.  Now 
as  there  are  four  of  these,  and  only  one  equation,  it  may  be  satisfied  in  an  infinite  variety  of  ways,  and  the 
problem  of  the  destruction  of  the  spherical  aberration  (as  it  is  termed)  becomes  indeterminate. 

The  equation  in  the  case  of  two  lenses  for  parallel  rays  is  312. 

General 


0  =      -^-  {(2  -  2  /«  +  /»)  R'*  +  (ft  +  2  /*  -  2  /*)  R'  R"  +  ft*  R"*  }  ;  («)          thTdestJc- 

"  tion  of  aber- 

/•                                                                                                                                                         -.  ration  in  a 

\(2  -  2  X''2  +  /»"3)  R"'2  +  0*''  +  2  p."*  —  2  /."")  R'"  Riv  +  u."3  R'«  }  Double  ^ 

[                                                                                                                                                       t  for  parallel 


{ 


(4  +  3  pV  -  3  /."«)  R'"  +  (p."  +  3  /,"«)  R-  j  +          1      2  +  3 


rays. 


This  equation,  if  L/  and  L",  the  powers  of  the  separate  lenses,  be  assigned,  is  of  a  quadratic  form  in  either      313. 
R',  R'1,  R'",  or  Riv ;  it  will  therefore  depend  on  the  supposition  adopted  to  limit  the  problem,  whether  these  Another 
quantities   admit  real    corresponding   values.      Now   the    equations    L'  =   (//  —   1)   (R'  —  R")    and   L"  =  form  °f  tn| 
0»"  -  1)  (R'"  -  R")  afford  the  means  of  eliminating  two  of  them,  and  the  resulting  equation  (in  R'  and  R'"  ^e  e1ua 
for  instance)  is 

0  =       L'  (   2  +/     R'*   - 


'  —  1 

T  "  C2+  A"  W,H        (  4  f/'  +  1)   _ ,         2  /'  +  I     T  „ 
(~~7~  ( S~  /'-I     L 

+  (;"  y,    4.   £v*    +  3^L.  t/u*.+ «±J2lL-i>, 

and,  as  the  unknown  quantities  R',  R"'  are  not  combined  by  multiplication,  the  equation  when  L'  and  L"  are 
given  is  of  an  ordinary  quadratic  form  with  respect  to  each.  This  equation  will  be  of  use  to  us  hereafter, 
when  we  come  to  treat  of  the  theory  of  refracting  telescopes. 

If  L/  and  I/'  be  not  given,  since  either  of  them  is  of  the  first  degree  in  terms  of  R',  R",  &c.,  the  equation      gjj 
(«)  is  of  the  third  degree  in  either  of  the  quantities  R',  R",  &c.,  or  in  L',  L",  if  either  R"or  Riv  be  elimi- 
nated.     Now  as  an  equation  of   the  third  degree  must  necessarily  have  at  least  one  real  root,  we  conclude, 
first,  that  in  a   double  lens,  if  the  curvatures  of  three  of  the  surfaces  be  given,  that  of  the  fourth  may    bt 
found,  so  as  to  destroy  the  spherical  aberration. 

Secondly.    That  if  the  curvature  of  one  surface  of  each   lens,  and  the   power  of  either,  or  that  of  the   two       315 
combined,  be  given,  the  power  of  the  other  may  be  found  so  as  to   destroy  the  spherical  aberration.      This  is 
evident ;    for,  supposing  R'  and  R'"  given,   and  either  L'  or  L/',  or  L'  +  L",  also  given,  the  equation  (i<) 
becomes   an    ordinary  cubic    in    which    I/  or  I/',  as  the  case  may  be,  is    the  only  unknown    quantity,    and 
therefore  necessarily  admits  a  real  value. 

As  examples  of  aplanatic  combinations,  we  may  set  down  the  following  cases,  in  which  a  lens  of  glass  of      315 
the  refraction  1.50,  and  of  the  best  form,  having  the  radii  of  its  surfaces  respectively  +  5.833  and  —  35.000 
inches,  and  its  focal  length  10.000    inches,   has  its    aberration  corrected  by  applying  behind  it  another  lens 
of  similar  glass,  as  in  fig.  55.     This  lens  is  a  meniscus.     If  its  curvatures  be  determined  by  the  condition  of  Fig.  55. 
giving  the    maximum  of  power  to  the  combination,  the  radii  of  its  surfaces  and  its  focal  length  will  be  as 
follows:  radius  of  first  surface,  =  +  2.054  inches  ;  radius  of  second  surface,  =  +  8.128;  focal  length  of  cor- 
recting lens,  =  +  5.497  ;   focal  length  of  the  two  combined,  =  -f  3.474.      On  the   other  hand,  if  we  deter- 
mine the  second  lens  by  the  condition  of  the  resulting  combination,  having  a  focal  length  as  nearly  10.000 
as  is  consistent  with  perfect  aplanaticity,  we  shall  find  radius  of  first  surface,  =  +  3.688 ;  radius  of  second, 
=  +  6.291  ;  focal  length  of  correcting  lens,  =  +  17.829  ;  focal  length  of  the  combination,  =  +  6.407. 

The  effect  of  aberration   may  be  very  prettily  exhibited  by  covering    a    large    convex    lens  with    a    paper      «*17 


392 


LIGHT. 


Light,  screen  full  of  small  round  holes,  regularly  disposed,  and,  exposing  it  to  the  sun,  receiving  the  converged  rays 
_  °  -i_'  on  a  white  paper  behind  the  lens,  which  should  be  first  placed  very  near  it,  and  then  gradually  withdrawn.  The 
pencils  which  pass  through  the  holes  will  form  spots  on  the  screen,  and  their  disposition  will  become  more  and 
more  unequal  over  the  surface,  as  the  screen  is  further  removed  ;  those  at  the  circumference  becoming  crowded 
together  before  the  central  ones.  The  manner  in  which  the  several  spots  corresponding  to  central  rays  blend 
together  into  one  image  at  the  focus,  and  those  formed  by  the  exterior  ones  are  scattered  round  it,  gives  us  a 
very  good  idea  of  the  variation  of  density  of  the  rays  in  the  circle  of  aberration  at  or  near  the  principal  focus; 
and  if  the  white  screen  be  waved  rapidly  to  and  fro  in  the  cone  of  rays,  so  as  to  pass  over  the  focus  at  each 
oscillation,  the  whole  cone  will  be  seen  as  a  solid  figure  in  the  air,  and  the  place  of  the  circle  of  least  aberra- 
tion will  become  evident  to  the  eye,  forming  altogether  a  very  pleasing  and  instructive  experiment. 


Part  I. 


§  XI.    Of  the  Foci  for  Oblique  Rays,  and  of  the  Formation  of  Images. 


318. 


Foci  of 
oblique 
pencils. 


We  have  hitherto  considered  rays  as  converging  to,  or  diverging  from,  a  single  point ;  but  as  this  is  not 
the  case  with  luminous  bodies  of  a  sensible  diameter,  we  now  proceed  to  examine  the  cases  of  refraction  at 
spherical  surfaces,  where  more  than  one  radiant  point  is  concerned,  or  where  several  pencils  are  incident  at 
once  on  the  surface.  We  shall  take  for  our  positive,  or  fundamental  case,  as  we  have  done  all  along,  that  of 
converging  rays  incident  on  the  convex  side  of  a  more  refractive  medium  than  the  ambient  one,  and  derive  till 
others  from  it  by  the  changes  in  the  sign  and  relative  magnitudes  of  R,  D,  &c. 

In  fig.  56,  then,  let  Q  and  Q'  be  the  foci  of  two  pencils  of  convergent  rays  incident  on  the  spherical  surface 
C  C',  whose  centre  is  E.  Draw  Q  E  C,  Q'  E  C',  cutting  the  surface  in  C  and  C',  and,  regarding  C  E  Q  as  the 

axis  of  the  pencil  R  Q,  S  Q,  T  Q,  the  focus  of  refracted  rays  will  be  found  by  taking  a,  such  as  that  ,    or 

Cq 

f,  shall  be  equal  to  (1  —  m)  R  +  wD,  (247,  e.)     Similarly,  regarding  C'E  Q'  as  the  axis  of  the  pencil  con- 
verging to  Q',  the  focus  q  will  be  had  by  the  equation 


T  =  f  -  (1  -  m)  R  +  m  D'. 

Thus  when  C'Q'  =  C  Q,  Cq'  will  also  equal  C  q,  and,  in  general,  when  the  locus  of  the  point  Q  is  given,  that 
of  q  may  be  found. 

Definition.  The  image  of  an  object,  in  Optics,  is  the  locus  of  the  focus  of  a  pencil  of  rays  diverging  from, 
Images  m  or  converging  to,  every  point  of  it,  and  received  on  a  refracting  surface.  Thus,  supposing  C  Q'  to  be  a  line, 
d  fined  or  sur^ace'  every  point  of  which  may  be  regarded  as  a  focus  of  incident  rays,  qq1  is  its  image. 

320           Problem.    To  find  the  form  of  the  im^ge  of  a  straight  line  formed  by  a  spherical  refracting  or  reflecting 
Form  of  the  surface.  


image  of  a 
straight  line 


Then  we  have 
and  therefore 

we  have,  consequently, 


1 


1  -m 


cv 


+ 


(1  —  m)  a'  +  mr  ' 


(1  —  m)  a'  +  mr 


m  r  (of  —  r) 
(1  —  m)  a'  -\-  mr  ' 


'    {(1  -  m)  a'  +  mr  )*   ' 
But,  by  similar  triangles,  E  q' :  E  M  :  :  E  Q'  :  E  Q,  or 

equating  these  two  values  we  get 

a  (1  —  TO)  of  +  m  r  _,  _      m  r  (a  —  x) 


m  r 


I  —  m 


.     so  that  eliminating  a',  by  substituting  this  value  for  it,  we  get  for  a  final  equation  between  *  and  y,  or  for  the 
wction°mC    eq«ati<>n  of  the  image 


which  belongs  to  a  conic  section. 

321.          Problem.    When  an   oblique  pencil   is   incident  on  any   system   of  spherical  surfaces,  to  find  the  foam  of 
refracted  rays. 


LIGHT.  393 

Light.  Take  E',  (fig.  57,)  the  centre  of  the  first  surface,  and  let  Q'  be  the  focus  of  incident  rays.     Join  Q'  E'  and     P«r*  !• 

produce  it  to  C',  then  will  C'  be  the  vertex  of  the  surface  corresponding  to   the  pencil  whose  focus  is  Q'  ;  and  *"••  ~^—  — 
takintr  Foci  of 


____ 

C'  Q"    "        C'  E'  C'  Q7  cident  on  a 

system  ot' 

Q"  will  be  the  focus  of  refracted  rays.      Again,  join  Q"  and  E",  the  centre  of  the  second  surface,  produce  spherical 
to  C",  and  take  %£%; 

1  1  —  m"  m" 

C"  Q"'  ~        C"  E"       "   C"  Q" 

and  Q'"  will  be  the  focus  after  refraction  at  the  second  surface,  and  so  on. 

Carol.  In  the  case  of  an  infinitely  thin  lens,  when  the  obliquity  is  small,  it  is  evident,  from  this  construction,      322. 
that  the  focus  of  oblique  rays  will  lie  at  the  same  distance  from  the  lens  with  that  of  rays  convergent  to,  or 
divergent  from,  a  point  in  the  axis  at  the  same  distance  with  the  focus  of  incident  rays,  but  instead  of  lying  in  the 
axis,  will  deviate  from  it. 

Definition.  The  centre  of  a  lens  is  a  point  in  its  axis  where  a  line  joining  the  extremities  of  two  parallel  radii       323. 
of  its  two  surfaces  cuts  the  axis.     Thus,  in  the  various  lenses  represented  in  fig.  58,  59,  60,  and  61,  E'  A.  and  E"  B  £*ntre  of  a 
being  two  parallel  radii  ;  join  B  A,  and  produce,  if  necessary,  till  it  meets  the  axis  in  X,  arid  X  is  the  centre. 

Carol.  1.    The  centre  is  a  fixed  point  ;  for,  since  A  E'  and  B  E''  are  parallel,  we  have  E'  X  :  E'  E"  :  :  A  E'  :       324. 
B  E''—  AE',  in  which  proportion  three  terms  being  invariable,  the  other  is  so  also. 

Coral.  2.  If  C'  C",  the  interval  of  the  surfaces  or  thickness  of  the  lens,  be  put  equal   to  t  (t  being  always       325. 
positive)  and  the  curvatures  be  respectively  R'  and  R",  we  have,  for  the  distance  of  the  centre  from  the  first 
surface,  or  for  C'  X,  the  following  value, 

R" 

c'x  =  -BTTWr-t- 

Coral.  3.    If  a  ray  be  so  incident  on  a  lens  that  its  direction  after  the  first  refraction  shall  pass  through  its      326. 
centre,  it  will  suffer  no  deviation.     This  is  evident,  because  its  course  within  the  lens  will  be  A  B,  and  the  radii  Rays 
E'  A  and  E''B  being  parallel,  the  internal   angles  of  incidence  on  the   surfaces  are  equal,  and,  therefore,  the  'hrongh  the 
angles  of  refraction  both  ways  out  of  the  lens  ;  consequently  the  two  portions  of  the  ray  without  the  lens  are  uncfeviatec? 
parallel 

Coral  4     If  the  thickness  of  a  lens  be  very  small,  the   ray  passing  through  its  centre  may  be  regarded   as       337. 
undergoing  no  refraction  whatever  ;   for  the  portion  A  B  within  the  lens  being  very  small,   the  two  portions 
exterior  to  the  lens  (being  parallel)  may  be  regarded  as  one  ray.     This  is,  a  fortiori,  still  nearer  the  truth  when 
the  obliquity  of  the  ray  to  the  axis  is  small  ;  because  then  the  portion  A  B  is  very  nearly  coincident  in  direction 
with  either  of  the  two  exterior  portions. 

Carol.  5.  Hence,  to  find  the  focus  of  refracted  rays  in  the  case  of  a  very  thin  lens  and  for  a  pencil  of  small      328. 
obliquity,  take  X,  the  centre  of  the  lens,  and  the  focus  will  lie  in  the  line  Q  X,  at  the  same  distance  from  the  lens  focus  of  a, 
as  if  the  axis  of  the  incident  pencil  were  coincident  with  that  of  the  lens. 

Proposition.  When  a  luminary,  or  illuminated  object,  is  placed  before  a  double  or  plano-convex,  or  meniscus  throu<*lfaC' 
lens,  at  a  distance  from  it  greater  than  its  focal  length,  there  will  be  formed  behind  the  lens  an  image,  similar  thin  lens. 
to  the  object,  but  inverted  ;  and  the  object  and  image  subtend  the  same  angle  at  the  centre  of  the  lens.  329. 

For  the  pencil  of  rays  which  emanates  (either  by  direct  radiation  or  by  reflexion)  from  any  point,  as  P,  of  the  Fig.  62. 
object,  will  after  refraction  be  all  made  to  converge  to  a  point  p  behind  the  lens,  or  at  least  very  nearly  so.  A"  lnverted 
Were  the  aberration  of  the  lens  evanescent,  the  convergence  would  be  mathematically  exact  ;  and  since,  when-  ^-^  °s  ! 
ever  the  aperture  of  the  lens  and  the  obliquity  of  the  pencil  are  small,  the  aberration  is  so  very  minute,  that  the  fornied 
space  over  which  the  rays  are  spread  may  be  regarded  as  a  physical  point,  and  every  physical  point  in  the  object  behind  a 
will  have  a  corresponding  point  in  the  image.     Now,  C  being  the  centre  of  the  lens,  the  line  joining  Pp  passes  convex  lens. 
through  C  ;  and  the  same  being  true  of  the  line  joining  any  other  corresponding  points  of  the  object  and  image, 
it  follows,  by  similar  triangles,  that  the  object  and  image  are  similar  in  figure  ;  and  as  the  rays  cross  at  C,  the 
image  is  inverted,  and  subtends  the  same  angle  p  C  q  at  C  that  the  object  does  on  the  other  side. 

If  a  screen  of  white  paper  be  placed  at  qp,  this  image  will  be  rendered  visible  as  a  picture  of  the  object.    The      330. 
experiment  may  be  tried  with  any  magnifier  or  spectacle-glass  at  a  window,  when  the  forms  of  external  objects,  Camera 
the  houses,  trees,  landscape,  &c.  will  be  painted  on  the  paper  screen  with  perfect  fidelity,  forming  a  miniature  of  obscura 
the  utmost  delicacy  and  beauty.     This  is  the  principle  of  the  common  camera  obscura,  in  which  the  rays  from  «xPlam«d- 
external  objects  are  thrown  by  an  inclined  looking-glass  downwards,  and  being  received  on  a  convex  lens,  are 
brought  to   their  focus  on  a  white  horizontal  table,  in  a  room  where  no  other  light  is  admitted.     On  this  table 
a  moving  picture  of  all  external  objects,  in  their  proper  forms,  colours,  and  motions,  is  seen,  infinitely  more  correct 
and  beautiful  than  the  most  elaborate  painting.     See  fig.  63,  in  which  P  is  the  object,  A  B  the  reflector,  B  C  the 
Jens,  and  p  the  image  on  the  table  D. 

If  the  rays,  instead  of  being  received  on  white  paper,  be  received  on  a  plate  of  glass  emeried  on  one  side,  331. 
the  picture  may  be  seen  by  an  eye  placed  at  the  other  side  of  the  glass,  as  well  as  by  one  in  front  of  it  ;  for  it  is 
a  property  of  such  roughened  transparent  surfaces  to  scatter  the  rays  which  fall  on  them,  not  only  by  reflexion 
outwards,  but  by  refraction  inwards.  If  the  surface  be  but  slightly  roughened,  however,  the  picture  will  appear 
much  less  vivid  when  looked  at  obliquely  than  when  the  eye  is  placed  immediately  behind  it  ;  and  in  thi» 
voi.  iv,  3  t 


394  LIGHT. 

Light.      latter  situation  the  emeried  glass  may  even  be  removed  altogether,  and  the  image  will  still  be  seen,  and  even  more      Part  I. 
*•"• ~v~™*/  distinctly,  as  if  a  real  object  stood  in  the  place  in  all  respects  similar  to  the  picture. 

We  may  examine  the  image  on  the  roughened  glass  with  a  magnifying  glass,  or  microscope.  It  will  then 
appear  as  a  delicate  painting,  accommodating  itself  to  all  the  inequalities  of  the  surface.  But  if,  in  the  act  of 
so  examining  it,  the  rough  glass  be  removed,  the  painting  remains  as  if  suspended  in  air,  and  the  objects  it 
represents  are  seen  brought  nearer  to  the  eye,  and  enlarged  in  their  dimensions.  In  short,  we  have  formed  a 
telescope. 

333.  If  *ne  lens  used  to  form  the  image  be  a  concave  one,  or  if  a  convex  reflector  be  used,  as  in  fig.  64  and  65, 
the  rays,  after  refraction  or  reflexion,  diverge,   not  from  any  actual  points  in  which  they  cross,  but  from  points 
in  which  they  would  cross  if  produced  backwards.     There  is  in  this  case,  then,  no  real  image  formed  capable 
of  being  received  on  a  screen,  but  what  is  called  a  virtual  one,  visible  to  the  eye  if  properly  situated,  either  un- 
assisted or  aided  by  a  magnifier,  and  situated  on  the  same  side  of  the  lens,  or  oil  the  contrary  side  of  the  reflector 
with  the  object,  and  therefore  erect. 

334.  The  perfection  of  the  image  formed  by  a  lens  or  reflector,  its  exact  resemblance  to  the  object,  and  the  distinct- 
ness of  its  parts,  will  depend  on  the  exact  convergence  of  all  the  rays  of  pencils  emanating  from  every  physical 
point  of  the  object  in  strict  mathematical  points,  or  in  as  near  an  approach  to  such  points  as  may  be.     If, 
therefore,  a  lens  of  considerable  diameter  be  used,  especially  if  the  curvatures  of  its  surfaces  be  improperly  chosen 
so  as  to  produce  much  aberration,  the  image  will  be  confused ;   for  each  point  of  the  object  will  form,  not  a 
point,  but  a  small  circular  spot  in  the  image,  over  which  the  rays  are  diffused  ;   and  as  these  spots  overlap  and 
encroach  on  each  other,  distinctness  is  destroyed.     For  the  formation,  therefore,  of  perfect  images,  the  destruc- 
tion of  aberration  is  the  essential  condition ;  and  whatever  imperfections,  either  in  the  figures  of  the  reflecting  or 
refracting  surfaces  used,  or  in  the  materials  of  which  they  are  composed,  tends  to  throw  the  rays  aside  from  their 
strict  geometrical  direction,  must,  of  course,  confound  the  images.     Hence,  in  the  formation  of  optical  images, 
there  are  three  great  points  to  be  attended  to  :  first,  perfect  polish  of  the  surfaces ;  secondly,  perfect  homogeneity 
in  the  material  employed ;    thirdly,  strict  conformity  in  the  figures  of  the  reflecting  and  refracting  surfaces  to 
geometrical  rules,  and  the  results  of  analysis. 

335.  There  is  one  case  where  the  aberrations  of  all  kinds  are  rigorously  destroyed,  and  in  which  the  image  is  perfect. 
It  is  when  the  rays  are  reflected  at  a  plane  surface.     For  (fig.  66)  if  P  Q  be  an   object   placed   before  a  plane 
reflector  AB,  and  if  perpendiculars  be  let  fall  from  every  point  of  the  object  to  the  surface,  and  on  the  other  side 
points  in  these  be  taken  at  the  same  distances  respectively  behind  the  surface  as  p,  q,  these  points  will  form  the 
image.     Now  we  have  seen,  that  all  rays  from  any  point  P,  reflected  at  A  B,  will  after  reflexion  diverge  strictly 
from  p  its  image.     Thus,  the  image  is  as  perfect  and  free  from  aberration  as  the  object ;  and  will  appear,  to  an 
eye  placed  so  as  to  receive  the  rays,  like  a  real  object  placed  behind  the  reflector. 

336.  Corel.  The  image  formed  by  a  plane  reflecting  surface  is  similar  and  equal  to  the  object,  and  any  correspond- 
ing lines  in  both  are  equally  inclined  to  the  reflecting  surface.     A  common  looking-glass  is  the  best  illustration 
of  this  case. 

337.  Proposition.    To  determine  the  image  of  any  object  formed  by  a  plane  refracting  surface.     Let  B  C  be  the 
surface,  P  Q  the  object.     From  any  point  Q  draw  Q  C  perpendicular  to  the  surface,  and,  ft  being  the  index  of 
refraction,  if  we  regard  the  surface  as  a  sphere  of  infinite  radius,  we  have  R  its  curvature  =  0,  and  the  equation 

/=(!--  m)  R  +  mD  becomes  simply  /  =  m  D.     Now  /  =  — —  ;    D  =  ;    and  m  —   — .      Hence 

O  q  *-*  >K  /* 

this  equation,  translated  into  geometrical  language,  gives  C  q  —  fi  X  C  Q. 

338.  In  the  case  represented  in  the  figure,  the  refraction  is  made  out  of  a  denser  medium   into  a  rarer,  the   object 
being  immersed  in  the  denser  (as  under  water),  and  the  eye  of  a  spectator  in  the  rarer  (as  in  air)  :   the  image  q  of 
the  point  Q  is  therefore  nearer  the  surface  than  Q,  (because  in  this  case  /t  is  less  than  unity.)     The  same  holds 
good  of  all  other  points  of  the  image  ;  so  that  the  whole  object  will  appear  raised  by  refraction,  as  in  the  familiar 
experiment  where  a  shilling  is  laid  in  an  empty  vessel,  and  the  eye  withdrawn  till  the  shilling  is  hidden  by  the 
edge,  but  reappears  again,  as  if  raised  up,  when  the  vessel   is  filled  with  water.      On  the  other  hand,  to  an 
eye  placed  under  water,  external  objects  would  appear  farther  removed  by  the  effect  of  refraction. 

339.  Carol.  1.  The  image  of  a  straight  line  PQ   in  the  object  is  a  straight  line  pq  in  the  image,  less  inclined  to 
the  surface  if  the  refraction  be  made  from  a  denser  into   a  rarer  medium.     Thus,  if  a  stick  D  A  P  Q   be  partly 
plunged  into  water,  the  immersed  portion  AQ  forms  the  image  A  q  less  inclined;  so  that  to  a  spectator  in  air, 
the  stick  appears  broken  and  bent  upwards  at  A.     The  appearance  is  familiar  to  every  one. 

340.  In  refraction  at  a  plane  surface,  however,  the  rays  do  not  rigorously  diverge  from,  or  converge  to,  a  single 
point.     Therefore  the  above  result  is  only  approximately  correct,  and  supposes  the  rays  to  be  incident  nearly  at 
right  angles  to  the  surface.     And  this  leads  us  to  the  consideration  of  oblique  vision  through  refracting  surfaces, 
or  in  reflectors  of  any  figure. 

341.  The  eye  sees  by  the  rays  which  enter  it,  and  judges  of  the  existence  of  an  object,  by  the  fact  of  rays  diverging 
Oblique       sensibly  from  some  point  in  space.     If,  then,  rays  diverge  rigorously  from  a  point,  the  eye  which  receives  them 

is  irresistibly  led  to  the  belief  (unless  corrected  by  experience  and  judgment)  of  an  object  being  there  ;  the 
fractmg  or"  illusion  >s  complete,  and  vision  perfect.  But  if  such  divergence  be  only  approximate,  as  when  the  density  of  the 
reflecting'  rays  wmcn  reach  the  eye  in  any  one  direction  is  very  much  greater  than  in  directions  adjacent  on  either  side, 
surfaces  of  vision  is  still  produced,  only  less  distinct,  in  proportion  to  the  degree  of  deviation  from  strict  mathematical 
»ny  figure,  divergence  of  the  rays  which  produce  it.  Suppose,  now,  Q  to  be  a  radiant  point  placed  anywhere  with  respect 
F'S  68.  (.o  the  refracting  or  reflecting  surface  A  C  B,  (fig.  68,)  and  let  A  q  F  B  be  the  caustic  formed  by  the  intersection  of 

all  the  refracted  or  reflected  rays.     Let  us  suppose  an  eye  placed  at  E,  and  from  thence  draw  E  q  a  tangent 


LIGHT.  395 

Light,  to  the  caustic,  which  continue  to  the  surface  C,  and  join  Q  C.  Then  it  is  obvious,  that  any  small  pencil  Q  C,  Q  C  P«t  I. 
1  diverging  from  Q,  will  form  a  focus  at  q  (Art.  134,  &c.)  from  which  it  will  afterwards  diverge,  and  fall  on  the  eye  ' 
at  E,  nearly  as  if  the  rays  came  from  a  mathematical  point;  and  from  what  was  said  in  Art.  161  and  163,  it 
appears  that  the  density  of  rays  in  the  cone  q  E  is  infinitely  greater  than  in  any  adjacent  cone  having  the  eye  for 
its  base ;  so  that  q  will  appear  as  an  image  of  Q,  more  or  less  confused,  in  proportion  to  the  degree  of  curvature 
of  the  caustic  at  q  ;  for  it  is  evident,  that  if  the  curvature  be  great,  the  assumed  concentration  of  any  small  finite 
pencil  Q  C  C'  in  one  mathematical  point  q,  will  deviate  more  from  truth  than  if  the  caustic  approach  nearly  to  a 
straight  line. 

Carol.    As  the  eye  shifts  its  place,  the  apparent  position  of  an  object  seen  in  a  reflecting  or  refracting  surface       342. 
shifts  also,  for  as  E  varies,  the  tangent  E  q  shifts  its  place  on  the  caustic,  and  the  point  of  contact  q,  or  the 
place  of  the  image  shifts. 

This  doctrine  may  be  illustrated  by  a  very  familiar  instance.     If  we  look  through  a  surface  of  still  water,  not       343. 
very  deep,  but  having  a  level  horizontal  bottom,  the  bottom  will  not  appear  a  plane,  but  will  seem  to  rise  on  all  Apparent 
sides,  and  approach  nearer  the   surface  the  more  obliquely  we  look.     To  explain  this,  let  Q  be  a  point  in  the  figu.re  of  thc 
bottom,  and  let  QPe  be  the  course  of  the  pencil  of  rays  by  which  an  eye  at  e  sees  it  (fig.  39)  on  the  visual  ray.  Jj°™ntai 
The  point  in  the  caustic  to  which  e  P  produced  is  a  tangent,  is  Y ;  and  from  the  form  of  the  caustic  D  Y  B  (see  stjiuTater 
Art.  238)  it  is  obvious,  that  Y  is  nearer  the  surface  the  more  oblique  e  P  is  to  it.     The  apparent  figure  of  the  Fig.  39. 
bottom  will  therefore  be  thus  determined.     From  the  eye  E  (fig.  69)  draw  any  line  E  g  to  the  point  G  of  the  Fig.  69. 
surface;   and  having  drawn  P  Y  parallel  to  E  G,  touching  the  branch  1)  Y  B  of  the  caustic  having  Q,  vertically 
below  E  for  a  radiant  point  in  Y,  prolong  E  G  to  H,  making  G  H  =  P  Y,  then  will  H  be  the  image  of  the  point 
Q'  in  the  bottom,  belonging  to  the  caustic  D'  H  B' ;  and  the  locus  of  H,  or  the  apparent  form  of  the  bottom, 
will   be  the  curve  D  F  H,  having  a  basin-shaped  curvature  at  D,  a  point  of  contrary  flexure   at  F,  and  an 
asymptote  C  G  K  coinciding  with  the  surface. 

But,  to  return  to  the  case  of  images  formed  by  rays  incident  at  very  small   obliquities   and  nearly  central,       344. 
the  following  rules  for  determining   their  places,  magnitudes,  and  apparent  situations  in  all  cases  of  spherical  Rules  for 
surfaces,  will  be  convenient  to  bear  in  memory,  and  will  need  no  express  demonstration  to  the  reader  of  the  fore-  fincling  'he 
going  pages. 

Rule  1.  Any  image  formed,  or  about  to  be  formed,  by  converging  rays,  or  from  which  rays  diverge,  may  be       345*^ 
regarded  as  an  object. 

Rule  2.  In  spherical  reflectors  the  object  and  its  image  lie  on  the  same  side  of  tb*  principal  focus.  They  move  346. 
in  contrary  directions,  and  meet  at  the  centre  and  surface  of  the  reflector.  The  Jjstance  of  the  image  from  the  Rule  fot 
principal  focus  and  centre  is  had  by  the  proportion  reflectors. 

QF:FE  ::EF:F9:  :  QE  :  E  9, 

and  the  image  is  erect  when  the  object  and  surface  lie  on  the  same  side  of  the  principal  focus  ;  but  inverted  when 
on  contrary  sides.  The  relative  magnitudes  of  the  object  and  image  (being  as  their  distances  from  the  centre) 
are  given  by  the  proportion 

object  :  image  :  :  Q  F  :  F  E  :  :  distance  of  the  object  from  the  principal  focus  :  focal  length  of  reflector. 

Rule  3.  In  thin  lenses,  of  all  species,  if  Q  be  the  place  of  the  object,  q  of  its  image,  E  the  centre  of  the  lens,       347 
F  the  principal  focus  of  rays  incident  in  a  contrary  direction,  then  will  the  object  and  image  lie  on  the  same,  or  Rule  for 
opposite  side  of  the  lens,  according  as  the  object  and  lens  lie  on  the  same  or  opposite   sides  of  the  principal  'enses- 
focus  F.     In  the  former  case  the  image  is  erect,  in  the  latter  inverted,  with  respect  to  the  object.     The  distance 
of  the  image  from  the  lens,  or  from  the  object,  is  had  by  the  proportions 

QF  :  FE  :  :  QE  :  Eq;         QF  :  QE  :  :  QE  :  Q  q; 

and  the  magnitude  of  the  object  is  to  that  of  the  image  as  the  distance  of  the  object  from  F  is  to  the  focal  length, 
or  as  Q  F  :  F  E. 

Rule  4.  In  all  combinations  of  reflectors  and  lenses,  the  image  formed  by  one  is  to  be  regarded  as  the  object,  343. 
whose  image  is  to  be  formed  by  the  next,  and  so  on,  till  we  come  to  the  last. 

It  has  been  already  remarked  (Art.  6)  that  visible  objects  are  distinguished  from  optical  images  by  this,  that  349. 
from  the  former  light  emanates  in  all  directions,  whereas  in  the  latter  it  emanates  only  in  certain  directions. 
This  is  an  important  limitation  in  practical  optics.  A  real  object  can  be  seen  whenever  nothing  opaque  is 
interposed  between  it  and  the  eye.  An  image  can  only  be  seen  when  the  eye  is  placed  in  the  pencil  of  rays 
which  goes  to  form  it,  or  diverges  from  it.  Thus  in  the  case  represented  in  fig.  62,  except  the  eye  be  placed 
somewhere  in  the  space  D  q  p  H,  it  will  see  no  part  of  the  image,  EqD  and  A;;H  being  the  extreme  rays 
refracted  by  the  lens  from  the  extremities  of  the  object; 

The  brightness  of  an  image  is,  of  course,  proportional  to  the  quantity  of  light  which  is  concentrated  in  each  Brighlness 
point  of  it ;  and,  therefore,  supposing  no  aberration,  as  the  apparent  magnitude  of  the  lens  or  mirror  which  forms  of  images. 

it,  seen  from  the  object  x  -       ~p~      ~'     Or>  since  the  area  of  tne  obJect :  tnat  of  tne  image  :  :  (distance)" 

* o 
of  object  from  lens  :  (distance)2  of  image  ;  and  since  the  apparent  magnitude  of  the  lens  seen  from  the  object 

is  as  its    f-jj-7  ' — r: —  )  ,  the  brightness  or  degree  of  illumination  of  the  image  is  as  the  apparent 

3F2 


396 


LIGHT. 


Light.      magnitude  of  the  lens  seen  from  the  image,  alone,  whatever  be  the  distance  of  the  object.     Now  the  apparent 

— -v^**  magnitude  of  the  lens  seen  from  the  image  is  always  much  less  than  a  hemisphere.     Therefore  (even  supposing 

no  light  lost  by  reflection  or  refraction)  the  illumination  of  the  image  is  always  much  less  than  that  of  the  object. 

This  is  the  case  when  the  image  is  received  on  a  screen  which  reflects  all  the  rays,  or  when  viewed  by  an 

eye  behind  it  having  a  pupil   large  enough  to  receive  all  the  rays  which  have  crossed  at  the  image,  a  fortiori, 

then,  when  the  eye  does  not  receive  all  the  rays,  must  the  apparent  intrinsic  brightness  be  less  than  that  of  the 

object.     This  supposes  the  object  to  have  a  sensible  magnitude ;  but  when  both  the  object  and  its  image  are 

Images  are    physical  points,  the  eye  judges  only  of  absolute  light ;  and  the  light  of  the  image  is  therefore  proportional  to  the 

apparent  magnitude  of  the  lens,  as  seen  from  the  object.     In  the  case  of  a  star,  for  instance,  whose  distance  is 

their  objects  constant,  the  absolute  light  of  the  image  is  simply  as  the  square  of  the  aperture,  and  this  is  the  reason  why  stars 

can  be  seen  in  large  telescopes  which  are  too  faint  to  be  seen  in  small  ones. 


Part  L 


§  XII.  Of  the  Structure  of  the  Eye,  and  of  Vision. 


350. 


Description 
of  the  eye. 

Fig.  70. 

Aqueous 
humour. 
Its  compo- 
sition. 
Refractive 
power. 
Cornea. 
Its  figure 
an  ellipsoid 
of  revolu- 
tion. 


351. 

Iris.     , 


352. 

Crystalline. 
Its  figure. 


Refraction. 
Non-coinci- 
dence of  the 
axes  of  its 
surfaces. 


It  is  by  means  of  optical  images  that  vision  is  performed,  that  we  see.  The  eye  is  an  assemblage  of  lenses 
which  concentrate  the  rays  emanating  from  each  point  of  external  objects  on  a  delicate  tissue  of  nerves,  called 
the  retina,  there  forming  an  image,  or  exact  representation  of  every  object,  which  is  the  thing  immediately  per- 
ceived or  felt  by  the  retina. 

Fig.  70  is  a  section  of  the  human  eye  through  its  axis  in  a  horizontal  plane.  Its  figure  is,  generally  speaking, 
spherical,  but  considerably  more  prominent  in  front.  It  consists  of  three  principal  chambers,  filled  with  media 
of  perfect  transparency  and  of  refractive  powers,  differing  sensibly  inter  se,  but  none  of  them  greatly  different  from 
that  of  pure  water.  The  first  of  these  media,  A,  occupying  the  anterior  chamber,  is  called  the  aqueous  humour, 
and  consists,  in  fact,  chiefly  of  pure  water,  holding  a  little  muriate  of  soda  and  gelatine  in  solution,  with  a  trace 
of  albumen ;  the  whole  not  exceeding  eight  per  cent.*  Its  refractive  index,  according  to  the  experiments  of 
M.  Chossat,  t  and  those  of  Dr.  Brewster  and  Dr.  Gordon,}  is  almost  precisely  that  of  water,  viz.  1.337,  that  of 
water  being  1.336.  The  cell  in  which  it  is  contained  is  bounded,  on  its  anterior  side,  by  a  strong,  horny,  and 
delicately  transparent  coat  a,  called  the  cornea,  the  figure  of  which,  according  to  the  delicate  experiments  and 
measures  of  M.  Chossat,  §  is  an  ellipsoid  of  revolution  about  the  major  axis  ;  this  axis,  of  course,  determines  the 
axis  of  the  eye;  but  it  is  remarkable,  that  in  the  eyes  of  oxen,  measured  by  M.  Chossat,  its  vertex  was  never 
found  to  be  coincident  with  the  central  point  of  the  aperture  of  the  cornea,  but  to  lie  always  about  10° 
(reckoned  on  the  surface)  inwardly,  or  towards  the  nose,  in  a  horizontal  plane.  The  ratio  of  the  semi-axis 
of  this  ellipse  to  the  excentricity,  he  determines  at  1.3;  and  this  being  nearly  the  same  with  1.337,  the  index 
of  refraction,  it  is  evident,  from  what  was  demonstrated  in  Art.  236,  that  parallel  rays  incident  on  the  cornea  in 
the  direction  of  its  axis,  will  be  made  to  converge  to  a  focus  situated  behind  it,  almost  with  mathematical 
exactness,  the  aberration  which  would  have  subsisted,  had  the  external  surface  a  spherical  figure,  being  almost 
completely  destroyed. 

The  posterior  surface  of  the  chamber  A  of  the  aqueous  humour  is  limited  by  the  iris  /)  <y,  which  is  a  kind  of 
circular  opaque  screen,  or  diaphragm,  consisting  of  muscular  fibres,  by  whose  contraction  or  expansion  an 
aperture  in  its  centre,  called  the  pupil,  is  diminished  or  dilated,  according  to  the  intensity  of  the  light.  In  very 
strong  lights  the  opening  of  the  pupil  is  greatly  contracted,  so  as  not  to  exceed  twelve  hundredths  of  an  inch  in 
the  human  eye,  while  in  feebler  illuminations  it  dilates  to  an  opening  not  exceeding  twenty-five  hundredths,||  or 
double  its  former  diameter.  The  use  of  this  is  evidently  to  moderate  and  equalize  the  illumination  of  the 
image  on  the  retina,  which  might  otherwise  injure  its  sensibility.  In  animals  (as  the  cat)  which  see  well  in 
the  dark,  the  pupil  is  almost  totally  closed  in  the  daytime,  and  reduced  to  a  very  narrow  line ;  but  in  the  human 
eye,  the  form  of  the  aperture  is  always  circular.  The  contraction  of  the  pupil  is  involuntary,  and  takes  place 
by  the  effect  of  the  stimulus  of  the  light  itself;  a  beautiful  piece  of  self-adjusting  mechanism,  the  play  of  which 
may  be  easily  seen  by  approaching  a  candle  to  the  eye  while  directed  to  its  own  image  in  a  looking-glass. 

Immediately  behind  the  opening  of  the  iris  lies  the  crystalline  lens,  B,  enclosed  in  its  capsule,  which  forms  the 
posterior  boundary  of  the  chamber  A.  Its  figure  is  a  solid  of  revolution,  having  its  anterior  surface  much  less 
curved  than  the  posterior.  Both  surfaces,  according  to  M.  Chossat,  are  ellipsoids  of  revolution  about  their 
lesser  axes ;  but  it  would  seem  from  his  measures,  that  the  axes  of  the  two  surfaces  are  neither  exactly  coincident 
in  direction  with  each  other,  nor  with  that  of  the  cornea.  This  deviation  would  be  fatal  to  distinct  vision 
were  the  crystalline  lens  very  much  denser  than  the  others,  or  were  the  whole  refraction  performed  by  it.  This, 
however,  is  not  the  case;  for  the  mean  refractive  index  of  this  lens  is  only  1.384,  while  that  of  the  aqueous 
humour,  as  we  have  seen,  is  1.337  ;  and  that  of  the  vitreous  C,  which  occupies  the  third  chamber,  is  1.339  ;  so 
that  the  whole  amount  of  bending  which  the  rays  undergo  at  the  surface  of  the  crystalline  is  small,  in  compa- 
rison with  the  inclination  of  the  surface  at  the  point  where  the  bending  takes  place,  and,  since  near  the  vertex,  a 


'  Chenevix,  Philosophical  Transaction*,  vol.  xciii.  p.  195. 

f  Bulletin  de  la  Sac.  Philomatique,  1818,  p.  94. 

J  Edinburgh  Philiaophical  Journal,  vol.  i.  p.  42. 

§  Sur  la  Covrburc  des  Milieux  Rifringeiu  ile  tCEil  chez  le  Baeuf.   Annalet  de  C/iim.  x.  p.  337. 

||  Dr.  Young's  Lectures  on  the  Mechanism  of  the  Eye,  Philosophical  Trantaclions,  vol.  xci. 


LIGHT.  S97 

Light,      material  deviation  in  the  direction  of  the  axis  can  produce  but  a  very  minute  change  in  the   inclination   of  the      P"t  I. 
—- Y™™*'  ray  to  the    surface,  this  cause   of  error  is  so  weakened  in   its  effect,  as,  probably,  to  produce  no  appreciable  »—»-v—— ' 
aberration.  Why not 

The  crystalline  is  composed  of  a  much  larger  proportion  of  albumen  and  gelatine  than  the  other  humours  of  "|Js""n° 
the  eye,  so  much  so  as  to  be  entirely  coagulable  by  the  heat  of  boiling  water.     It  is   somewhat  denser  towards       353 
the  centre  than  at  the  outside.     According  to  Dr.  Brewster  and  Dr.  Gordon,  the  refractive  indices  of  its  centre    Composi- 
middle  of  its  thickness,  from  the   centre  to  the  outside,  and  the  outside   itself,  are  respectively  1.3999,  1.3786,  tion  of crys- 
and  1.3767,  that  of  pure  water  being  1.3358.     This  increase   of  density  is  obviously  useful   in   correcting  the  talline- 
aberration,  by  shortening  the  focus  of   rays  near  the   centre,  according  to  the  rule  laid  down  in  Art.  299  for  j^^rds 
finding  the  effect  of  aberration.     The  effect  of  the  elliptic  figure  of  the  surfaces  is,  however,  a  matter  of  pretty  centre 
complex  calculation,  and  cannot  be  entered  upon  in  the  limits  of  this  essay.     Its  use  is,  probably,  to  correct  the 
aberration  of  oblique  pencils. 

The  posterior  chamber  C  of  the  eye  is  filled  with  the  vitreous  humour,  a  fluid  differing  (according  to  Chenevix)      354. 
neither  in  specific  gravity  nor  in  chemical  composition  in  any  sensible  respect  from  the  aqueous ;  and,  as  we  have 
already  seen,  having  a  refractive  index  but  very  little  superior  to  it. 

The  refractive  density  of  the  crystalline  being  superior  to  that  of  either  the  aqueous  or  vitreous  humour,  the      3^^ 
rays  which  are  incident  on  it  in  a  state  of  convergence  from  the  cornea,  are  made  to  converge  more,  and  exactly  Retina 
in  their  final  focus  is  the  posterior  surface  of  the  cell  of  the  vitreous  humour  covered  by  the  retina  d,  a  network 
(as  its  name  imports)  of  inconceivably  delicate  nerves,  all  branching  from  one  great  nerve  O,  called  the  optic 
nerve,  which  enters  the  eye  obliquely  at  the  inner  side  of  the  orbit,  next  the  nose.     The  retina  lines  the  whole 
of  the  cavity  C  up  to  i,  where  the  capsule  of  the  crystalline   commences.      Its  nerves  are  in  contact  with,  or 
immersed  in,   the  pigmenlum  nigruni,   a  very  black  velvety  matter,  which  covers  the  choroid  membrane  g,  and 
whose  office  is  to  absorb  and  stifle  all  the  light  which  enters  the  eye  as  soon  as  it  has  done  its  office  of  exciting 
the  retina ;  thus  preventing  internal  reflexions,  and  consequent  confusion  of  vision.     The  whole  of  these  humours 
and  membranes  are  contained  in  a  thick  tough  coat,  called  the  sclerotica,  which  unites  with  the  cornea,  and  forms  Sclerotica. 
what  is  commonly  called  the  white  of  the  eye. 

Such  is  the  structure  by  which  parallel  rays,  or  those  emanating  from  very  distant  objects,  are  brought  to  a  356. 
focus  on  the  retina.  But  as  we  require  to  see  objects  near,  as  well  as  at  a  distance,  and  as  the  focus  of  a  lens  phatle<- .of 
or  system  of  lenses  for  near  objects  is  longer  than  for  distant  ones,  it  is  evident  that  a  power  of  adjustment  must  forbear  ' 
reside  somewhere  in  the  eye ;  by  which  either  the  retina  can  be  removed  farther  from  the  cornea,  and  the  eye  objects. 
lengthened  in  the  direction  of  its  axis,  or  the  curvature  of  the  lenses  themselves  altered  so  as  to  give  greater 
convergency  to  the  rays.  We  know  that  such  a  power  exists,  and  can  be  called  into  action  by  a  voluntary  effort; 
and,  evidently,  by  a  muscular  action,  producing  fatigue  if  long  continued,  and  not  capable  of  being  strained 
beyond  a  certain  point.  Anatomists,  however,  as  well  as  theoretical  opticians,  differ  as  to  the  mechanism  by 
which  this  is  effected.  Some  assert,  that  the  action  of  the  muscles  which  move  the  eye  in  its  orbit,  called 
the  recti,  or  straight)  muscles,  when  all  contracted  at  once,  producing  a  pressure  on  the  fluids  within,  forces 
out  the  cornea,  rendering  it  at  once  more  convex,  and  more  distant  from  the  retina.  This  opinion,  however, 
which  has  been  advocated  by  Dr.  Olbers,  and  even  attempted  to  be  made  a  matter  of  ocular  demonstration  by 
Ramsden  and  Sir  E.  Home,  has  been  combated  by  Dr.  Young,  by  experiments  which  show,  at  least,  very 
decisively,  that  the  increase  of  convexity  in  the  cornea  has  little  if  any  share  in  producing  the  effect.  An  elon- 
gation of  the  whole  eye,  spherical  as  it  is  and  full  of  fluid,  to  the  considerable  extent  required,  is  difficult  to 
conceive  as  the  result  of  any  pressure  which  could  be  safely  applied,  as  to  give  distinct  vision  at  the  distance  of 
three  inches  from  the  eye,  (the  nearest  at  which  ordinary  eyes  can  see  well,)  the  sphere  must  be  reduced  to  an 
ellipsoid,  having  its  axis  nearly  one-seventh  longer  than  in  its  natural  state ;  and  the  extension  of  the 
sclerotica  thus  produced,  would  hardly  seem  compatible  with  its  great  strength  and  toughness.  Another  opinion, 
which  has  been  defended  with  considerable  success  by  the  excellent  philosopher  last  named,  is,  that  the  crystalline 
itself  is  susceptible  of  a  change  of  figure,  and  becomes  more  convex  when  the  eye  adapts  itself  to  near  distances. 
His  experiments,  on  persons  deprived  of  this  lens,  go  far  to  prove  the  total  want  of  a  power  to  change  the  focus 
of  the  eye  in  such  cases,  though  a  certain  degree  of  adaptation  is  obtained  by  the  contraction  of  the  iris,  which, 
limiting  the  diameter  of  the  pencil,  diminishes  the  space  on  the  retina  over  which  imperfectly  converged  rays  are 
diffused,  and  thus,  in  some  measure,  obviates  the  effect  of  their  insufficient  convergence.  When  we  consider 
that  the  crystalline  lens  has  actually  a  regular  fibrous  structure,  (as  may  be  seen  familiarly  on  tearing  to 
pieces  the  lens  of  a  boiled  fish's  eye,)  being  composed  of  layers  laid  over  each  other  like  the  coats  of  an 
onion,  and  each  layer  consisting  of  an  assemblage  of  fibres  proceeding  from  two  poles,  like  the  meridians 
of  a  globe,  the  axis  being  that  of  the  eye  itself;  we  have,  so  far  at  least,  satisfactory  evidence  of  a  muscular 
structure ;  and  were  it  not  so,  the  analogy  of  pellucid  animals,  in  which  no  muscular  fibres  can  be  discerned, 
and  which  yet  possess  the  power  of  motion  and  obedience  to  the  nervous  stimulus,  though  nerves  no  more 
than  muscles  can  be  seen  in  them,  would  render  the  idea  of  a  muscular  power  resident  in  the  crystalline 
easily  admissible,  though  nerves  have  as  yet  not  been  traced  into  it.  On  the  whole,  it  must  be  allowed,  that  the 
presumption  is  strongly  in  favour  of  this  mechanism,  though  the  other  causes  already  mentioned  may,  perhaps, 
conspire  to  a  certain  extent  in  producing  the  effect,  and  though  the  subject  must  be  regarded  as  still  open 
to  fuller  demonstration.  It  is  the  boast  of  science  to  have  been  able  to  trace  so  far  the  refined  contrivances 
of  this  most  admirable  organ ;  not  its  shame  to  find  something  still  concealed  from  its  scrutiny ;  for,  how- 
ever anatomists  may  differ  on  points  of  structure,  or  physiologists  dispute  on  modes  of  action,  there  is  that 
in  what  we  do  understand  of  the  formation  of  the  eye  so  similar,  and  yet  so  infinitely  superior,  to  a  product 
of  human  ingenuity, — such  thought,  such  care,  such  refinement,  such  advantage  taken  of  the  properties  of 
natural  agents  used  as  mere  instruments,  for  accomplishing  a  giv«-n  end,  as  force  upon  us  a  conviction  of 


398  LIGHT. 

Light.      deliberate  choice  and  premeditated  design,  more  strongly,  perhaps,  than  any  single  contrivance  to  be  found,     Part  t. 
— "V""''  whether  in  art  or  nature,  and  render  its  study  an  object  of  the  deepest  interest.  ^~"Y~ 

357.  The  images  of  external  objects  are  of  course  formed  inverted  on  the  retina,  and  may  be  seen  there,  by  dissect- 

Image  on  ;ng  off  the  posterior  coats  of  the  eye  of  a  newly-killed  animal,  and  exposing  the  retina  and  choroid  membrane 
the  retina  from  behind,  like  the  image  on  a  screen  of  rough  glass,  mentioned  in  Art.  331.  It  is  this  image,  and  this  only, 
diate™obfect  which  is  fdt  by  the  nerves  of  the  retina,  on  which  the  rays  of  light  act  as  a  stimulus ;  and  the  impressions 
of  vision.  therein  produced  are  thence  conveyed  along  the  optic  nerves  to  the  sensorium,  in  a  manner  which  we  must  rank  at 
present  among  the  profounder  mysteries  of  physiology,  but  which  appears  to  differ  in  no  respect  from  that  in 
which  the  impressions  of  the  other  senses  are  transmitted.  Thus,  a  paralysis  of  the  optic  nerve  produces,  while 
it  lasts,  total  blindness,  though  the  eye  remains  open,  and  the  lenses  retain  their  transparency ;  and  some  very 
curious  cases  of  half  blindness  have  been  successfully  referred  to  an  affection  of  one  of  the  nerves  without  the 
other.*  On  the  other  hand,  while  the  nerves  retain  their  sensibility,  the  degree  of  perfection  of  vision  is  exactly 
commensurate  to  that  of  the  image  formed  on  the  retina.  In  cases  of  cataract,  where  the  crystalline  lens  loses 
its  transparency,  the  light  is  prevented  from  reaching  the  retina,  or  from  reaching  it  in  a  proper  state  of  regular 
concentration,  being  stopped,  confused,  and  scattered  by  the  opaque  or  semi-opaque  portions  it  encounters  in  its 
passage.  The  image,  in  consequence,  is  either  altogether  obliterated,  or  rendered  dim  and  indistinct ;  and  the 
progress  of  blindness  is  accordingly.  If  the  opaque  lens  be  extracted,  the  full  perception  of  light  returns  ;  but 
one  principal  instrument  for  producing  the  convergence  of  the  rays  being  removed,  the  image,  instead  of  being 
formed  on  the  retina,  is  formed  considerably  behind  it,  and  the  rays  being  received  in  their  unconverged  state  on 
it,  produce  no  regular  picture,  and  therefore  no  distinct  vision.  But  if  we  give  to  the  rays,  before  their  entry  into 
the  eye,  a  certain  proper  degree  of  convergence,  by  the  application  of  a  convex  lens,  so  as  to  render  the  remain- 
ing lenses  capable  of  finally  effecting  their  exact  convergence  on  the  retina,  restoration  of  distinct  vision  is  the 
immediate  result.  This  is  the  reason  why  persons  who  have  undergone  the  operation  for  the  cataract  (which 
consists  either  in  totally  removing,  or  in  putting  out  of  the  way  an  opaque  crystalline)  wear  spectacles  of 
comparatively  very  short  focus.  Such  glasses  perform  the  office  of  an  artificial  crystalline.  A  similar  imper- 
fection of  vision  to  that  produced  by  the  removal  of  the  crystalline,  is  the  ordinary  effect  of  old  age,  and  its 
remedy  is  the  same.  In  aged  persons  the  exterior  transparent  surface  of  the  eye,  called  the  cornea,  loses  some- 
what of  its  convexity,  and  becomes  flatter.  The  power  of  the  eye  is  therefore  diminished,  (Art.  248  and  255,) 
and  a  perfect  image  can  no  longer  be  formed  on  the  retina.  The  deficient  power  is  however  supplied  by  a 
convex  lens,  or  spectacle-glass,  (Art.  268,)  and  vision  rendered  perfect  or  materially  improved. 

358  Short-sighted  persons  have  their  eyes  too  convex,  and  this  defect  is,  like  the  other,  remediable  by  the  use  of 

proper  lenses  of  an  opposite  character.  There  are  cases,  however,  though  rare,  in  which  the  cornea  becomes  so 
very  prominent  as  to  render  it  impossible  to  apply  conveniently  a  lens  sufficiently  concave  to  counteract  its  action. 
Such  cases  would  be  accompanied  with  irremediable  blindness,  but  for  that  happy  boldness,  justifiable  only  by  the 
certainty  of  our  knowledge  of  the  true  nature  and  laws  of  vision,  which  in  such  a  case  has  suggested  the 
opening  of  the  eye  and  removal  of  the  crystalline  lens,  though  in  a  perfectly  sound  state. 

359.  But  these  are  not  the  only  cases  of  defective  vision  arising  from  the  structure  of  the  organ,  which  are  suscep- 

Malconfor-    tible  of  remedy.     Malconformations  of  the  cornea  are  much  more  common  than  is  generally  supposed,  and  few 
•nations  of    eves  are>  jn  fac{t  free  from  them.     They  may  be  detected  by  closing  one  eye,  and   directing  the  other  to  a  very 
the  cornea.    narroW)  well-defined  luminous  object,  not  too  bright,  (the  horns  of  the  moon,  when  a  slender  crescent,  only  two 
or  three  days  old,  are  very  proper  for  the  purpose,)  and  turning  the  head  about  in  farious  directions.     The  line 
will  be  doubled,  tripled,  or  multiplied,  or  variously  distorted  ;  and  careful  observation  of  its  appearances,  under 
different  circumstances,  will   lead  to  a  knowledge  of  the  peculiar  conformation  of  the  refracting  surfaces  of  the 
Remarkable  eye  which  causes  them,  and  may  suggest  their  proper  remedy.     A  remarkable  and   instructive  instance  of  the 
case,  sue-     kind   has  recently  been  adduced  by  Mr.  G.  B.  Airy,   (Transactions  of  the   Cambridge  Philosophical  Society,) 
cessfully       jn  tne  case  of  one  of  n;s  own  eves  .  which,  from  a  certain  defect  in  the  figure  of  its   lenses,  he  ascertained  to 
refract  the  rays  to  a  nearer  focus  in  a  vertical  than  in  a  horizontal  plane,  so  as  to  render  the  eye  utterly  useless. 
This,  it  is  obvious,  would  take  place  if  the  cornea,  instead  of  being  a  surface  of  revolution,  (in  which  the  curvature 
of  all  its  sections  through  the  axis  must  be  equal,)  were  of  some  other  form,  in  which  the  curvature  in  a  vertical 
plane  is  greater  than  in  a  horizontal.     It  is  obvious,  that  the  correction  of  such  a  defect  could   never  be  accom- 
plished by  the  use  of  spherical  lenses.     The  strict  method,  applicable  in  all  such  cases,  would  be  to  adapt  a  lens 
to  the   eye,  of  nearly  the  same  refractive  power,  and  having  its  surface  next  the  eye  an  exact  intaglio  fac-simile 
of  the  irregular  cornea,  while  the  external  should  be  exactly  spherical  of  the  same  general  convexity  as  the  cornea 
itself;  for  it  is  clear,  that  all  the  distortions  of  the  rays  at  the  posterior  surface  of  such  a  lens  would  be  exactly 
counteracted  by  the  equal  and  opposite  distortions  at  the  cornea  itself,  t     But  the  necessity  of  limiting  the  cor- 
recting lens  to  such  surfaces  as  can  be  truly  ground  in  glass,  to   render  it  of  any  real   and   everyday  use,   and 
which  surfaces  are  only  spheres,  planes,  and  cylinders,  suggested   to   Mr.  Airy  the   ingenious  idea  of  a  double 
concave  lens,  in  which  one   surface  should  be  spherical,  the  other  cylindrical.     The  use  of  the   spherical  surface 
was   to  correct   the   general  defect  of  a  too  convex  cornea.     That  of  the  cylindrical    may  be   thus    explained. 
Suppose  parallel  rays  incident  on  a  concave  cylindrical  surface,  A  B  C  D,  in  a  direction  perpendicular  to  its  axis, 
Fig  71.        as  in  fig.  71,  and  let  S  S'  P  Pf  Q  Q'  T  T',  be  any  laminar  pencil  of  them  contained  in  a  parallelepiped  infinitely 

•  Wollaston,  on  Semi -decimation  of  the  Optic  Nerves,  Philoiophical  Traiaactioiu,  1824. 

t  Should  any  very  bad  cases  of  irregular  cornea  be  found,  it  is  worthy  of  consideration,  whether  at  least  a  temporary  distinct  vision  could 
not  be  procured,  by  applying  in  contact  with  the  surface  of  the  eye  some  transparent  animal  jelly  contained  in  i.spherical  capsule  of  glass  ;  or 
whether  an  actual  mould  of  the  cornea  might  not  be  tak.eo,  .ind  impressed  on  some  transparent  medium.  The  operation  would,  of  course,  lie 
delicate,  but  certainly  less  so  than  that  of  cutting  open  a  living  eye,  and  taking  out  its  contents. 


LIGHT.  399 

thin,  and  having  its  sides  parallel  to  the  axis.  Any  of  the  rays  S  P,  S'  P1,  of  this  pencil  lying  in  a  plane  APS  Part  I. 
perpendicular  to  the  axis,  will  after  refraction  converge  to,  or  diverge  from,  a  point  X,  also  in  this  plane  ;  and,  ^— v^. 
therefore,  all  the  rays  incident  on  P  Q,  P'  Q',  will  after  refraction  have  for  their  focus  the  line  X  Y,  in  the  caustic 
surface  A  F  G  D,  and  the  principal  focus  of  the  cylinder  will  be  the  line  F  G,  whose  distance  from  the  vertex 
C  C'  of  the  surface,  or  F  C,  is  the  same  with  the  focal  length  of  a  spherical  surface,  formed  by  the  revolution  of 
A  B  about  the  axis  F  C.  Thus  we  see  that  a  cylindrical  lens  produces  no  convergency  or  divergency  in  parallel 
rays,  incidental  in  the  plane  of  its  axis ;  while  it  converges  or  diverges  rays  in  a  plane  at  right  angles  to  the 
axis,  as  a  spherical  surface  of  equal  curvature  would  do.  If  then  such  a  cylindrical  surface  be  conjoined  with 
a  spherical  one,  the  focus  of  the  spherical  surface  will  remain  unaltered  in  one  plane,  but  in  the  other  will  be 
changed  to  that  of  a  lens  formed  by  it,  and  a  spherical  surface  of  equal  curvature  with  the  cylinder.  Hence  by 
properly  placing  such  a  cylindro-spheric  lens  across  the  defective  eye,  its  defect  will  be  (approximately,  at  least) 
counteracted.  It  would  be  wrong  to  conclude  our  account  of  this  interesting  application  of  mathematical 
knowledge  to  the  increase  of  the  comforts  and  improvement  of  the  faculties  of  its  possessor,  in  other  than  his 
own  words.  "  After  some  ineffectual  applications  to  different  workmen,  I  at  last  procured  a  lens  to  these 
dimensions,*  from  an  artist  named  Fuller,  at  Ipswich.  It  satisfies  my  wishes  in  every  respect.  I  can  now  read 
the  smallest  print  at  a  considerable  distance  with  the  left"  (the  defective)  "  eye  as  well  as  with  the  right.  I 
have  found  that  vision  is  most  distinct  when  the  cylindrical  surface  is  turned  from  the  eye :  and  as,  when  the 
lens  is  distant  from  the  eye,  it  alters  the  apparent  figure  of  objects  by  refracting  differently  the  rays  in  different 
planes,  I  judged  it  proper  to  have  the  frame  of  my  spectacles  made  so  as  to  bring  the  glass  pretty  close  to 
the  eye.  With  these  precautions,  I  find  that  the  eye  which  I  once  feared  would  become  quite  useless,  can  be 
used  in  almost  every  respect  as  well  as  the  other." 

Blindness,  partial  or  total,  may  be  caused,  not  only  by  the  opacity  of  the  crystalline  lens,  but  of  any  other  350. 
part,  or  by,  anything  extraneous  to  the  materials  of  which  they  consist,  interposed  between  the  external  trans- 
parent surface  of  the  cornea  and  the  retina.  In  all  such  cases,  if  the  sensibility  of  the  nerve  be  uninjured,  the 
restoration  of  sight  is  never  to  be  despaired  of.  In  a  recent  most  remarkable  case,  operated  by  Mr.  Wardrop, 
and  by  him  recorded  in  the  Philosophical  Transactions  for  1826,  blindness  from  infancy,  accompanied  with 
complete  obliteration  of  the  pupil,  by  a  contraction  of  the  iris,  owing  to  an  unskilful  operation,  performed  at 
six  months  of  age,  was  removed,  and  perfect  sight  restored  after  a  lapse  of  forty-six  years,  by  a  simple  removal 
of  the  obstruction,  by  breaking  a  hole  through  the  closed  membrane.  The  details  of  this  case  are  in  the 
highest  degree  interesting,  but  we  must  refer  the  reader  to  the  volume  of  the  Philosophical  Transactions  cited  for 
the  account. 

As  we  have  two  eyes,  and  a  separate  image  of  every  external  object  is  formed  in  each,  it  may  be  asked,  why  do       361. 
we  not  see  double  ?  and  to  some,  the  question  has  appeared  to  present  much  difficulty.     To  us  it  appears,  that  we  Single 
might  with  equal  reason  ask,  why — having  two  hands,  and  five  fingers  on  each,  all  endowed  with  equal  sensi-  J^™.™ 
bility  of  touch   and  equal  aptitude  to  discern  objects  by  that  sense — we  do  not  fed  decuple  ?     The  answer  is  the 
same  in  both  cases  :  it  is  a  matter  of  habit.     Habit  alone  teaches  us  that  the   sensations  of  sight  correspond  to 
any  thing  external,  and  to  what  they  correspond.     An  object  (a  small  globe  or  wafer  suppose)  is  before  us  on  a 
table  ;  we  direct  our  eyes  to  it,  i.  e.  we  bring  its  images  on  both  retina?  to  those  parts  which  habit  has  ascer- 
tained to  be  the  most  sensible  and  best  situated  for  seeing  distinctly  ;  and  having  always  found  that  in  such 
circumstances  the  object  producing  the  sensation  is  one  and  the  same,  the  idea  of  unity  in  the   object  becomes 
irresistibly  associated  with  the  impression.     But  while  looking  at  the  globe,  squeeze  the  upper  part  of  one  eye  Double 
downwards,  by  pressing  on  the  eyelid  with  the  finger,  and  thereby  forcibly  throw  the  image  on  another  part  of  vision 
the  retina  of  that   eye,  and  double  vision  is  immediately   produced,  two   globes  or  two  wafers  being   distinctly  "''finally 
seen,  which  appear  to  recede  from  each  other  as  the  pressure  is  stronger,  and  approach,  and  finally  blend  into  " 
one  as   it  is  relieved.     The  same  effect  may  be  produced  without  pressure,  by   directing  the   eyes  to  a  point  Another 
nearer  to,  or  farther  from  them  than  the  wafer  ;  the  optic  axes  in  this   case  being  both  directed  away  from  the  method, 
object  seen.     When  the  eyes  are  in  a  state  of  perfect  rest,  their  axes  are  usually  parallel,  or  a  little  diverging. 
In  this  state  all  near  objects  are  seen  double;   but  the  slightest  effort  of  attention  causes  their  images  to  coalesce 
immediately.     Those  who  have  one  eye  distorted  by  a  biow,  see  double,  till  habit  has  taught  them  anew  to  see 
single,  though  the  distortion  of  the  optic  axis  subsists. 

The  case  is  exactly  the  same  with  the  sense  of  touch.     Lay  hands  on  the  globe,  and  handle  it.     It  is  one,       362. 
nothing  can  be  more  irresistible  than  this  conviction.     Place  it  between  the  first  and  second  fingers  of  the  right  Single 
hand  in  their  natural  position.     The  right  side  of  the  first  and  left  of  the  second  finger  feel  opposite  convexities  ;  ^{^  k'1 
but  as  habit  has  always  taught  us  that  two  convexities  so  felt  belong  to  one  and  the  same  spherical  surface,  we  certajn  case, 
never  hesitate  or  question  the  identity  of  the  globe,  or  the  unity  of  the  sensation.     Now  cross  the  two  fingers, 
bringing  the  second  over  the  first,  and  place  the  globe  on  the  table  in  the  fork  between  them,  so  as  to  feel  the  left 
side  of  the  globe  with  the  right  side  of  the  second  finger,  and  the  right  with  the  left  of  the  first.     In  this  state  of 
things  the  impression  is  equally  irresistible,  that  we  have  two  globes  in  contact  with  the  fingers,  especially  if  the 
the  eyes  be  shut,  and  the  fingers  placed  on  it  by  another  person.     A  pea  is  a  very  proper  object  for  this  experi- 
ment.    The  illusion  is  equally  strong  when  the  two  fore  fingers  of  both  hands  are  crossed,  and  the  pea  placed 
between  them. 

So  forcible  is  the  power  of  habit  in   producing  single  vision,  that  it  will  bring  the  two  images  to  apparent      363. 
coalescence,  when  the  rays  which  form  one  of  them  are  really  turned  far  aside  from  their  natural  course.     To  Force  of 

show  this,  place  a  candle  at  a  distance,  and  look  at  it  with  one  eye  (the  left  suppose)  naked,  the  other  having  llabit  in 

producing 

— — "••• ' single  vision 

•  Badius  of  the  spherical  surface  34  inches,  of  the  cylindrical  4i.  illustrated  by 

experiment 


400  LIGHT. 

Light  before  it  a  prism,  with  a  variable  refracting1  angle,  (an  instrument  to  be  described  hereafter,  see  INDEX,)  and,  first,  Part  t 
s— v-"~-'  let  the  angle  be  adjusted  to  zero,  then  will  the  prism  produce  no  deviation,  and  the  object  will  appear  single.  >— — v—t 
Now  vary  the  prism,  so  as  to  produce  a  deviation  of  "2°  or  3°  of  the  rays  in  a  horizontal  plane  to  the  right.  The 
candle  will  immediately  be  seen  double,  the  image  deviated  by  the  prism  being  seen  to  the  left  of  the  other  ;  but 
the  slightest  motion,  such  as  winking  with  the  eyelids,  blends  them  immediately  into  one.  Again,  vary  the  prism 
a  few  degrees  more  in  the  same  direction  ;  the  candle  will  again  be  doubled,  and  again  rendered  single  by  winking, 
and  directing  the  attention  more  strongly  to  it  ;  and  thus  may  the  optic  axes  be,  as  it  were,  inveigled  to  an 
inclination  of  20°  or  30°  to  each  other.  In  this  state  of  things,  if  a  second  candle  be  placed  exactly  in  the 
direction  of  the  deviated  image  of  the  first,  but  so  screened,  that  its  rays  shall  not  fall  on  the  left  eye,  and  the 
prism  be  then  suddenly  removed  in  the  act  of  winking,  the  two  candles  appear  as  one.  If  the  deviation  of  the 
image  seen  with  the  right  eye  be  made  to  the  apparent  right,  the  range  within  which  it  is  possible  to  bring  them 
to  coalesce  is  much  more  limited,  as  it  is  much  more  usual  for  us  to  direct  by  an  effort  the  optic  axes  towards, 
than  from  each  other.  If  the  deviation  be  made  but  a  very  little  out  of  the  horizontal  plane,  no  effort  will 
enable  us  to  correct  it.  It  is  probable  that  «>..ie  cases  of  squinting  might  be  cured  by  some  such  exercise  in 
the  art  of  directing  the  optic  axes,  if  continued  perseveringly. 

364.  Such  is,  undoubtedly,  a  sufficient  explanation  of  single  vision  with  two  eyes  ;  yet  Dr.  Wollaston  has  rendered 
A  further      jt  probable  that  a  physiological  cause  has  also  some  share  in  producing  the  effect,  and  that  a  semi-decussation  of 
si^le           t'le  °Pt'c  nerves  takes  place  immediately  on  their  quitting  the  brain,  half  of  each  nerve  going  to  each  eye,  the 
vision.          right  half  of  each  retina  consisting  wholly  of  fibres  of  one   nerve,  and  the  left  wholly  of  the   other,  so  that  all 
Nervous       images  of  objects  out  of  the  optic  axis  are  perceived  by  one  and  the  same  nerve  in  both  eyes,  and  thus  a  power- 
sympathy,     ful  sympathy  and  perfect  unison  kept  up  between  them,  independent  of  the  mere  influence  of  habit.     Immediately 

in  the  optic  axis,  it  is  probable,  that  the  fibres  of  both   nerves  are   commingled,  and  this  may  account  for  the 
greater  acuteness  and  certainty  of  vision  in  this  part  of  the  eye. 

365.  Another  point,  on  which  much  more  discussion  has  been  expended  than  it  deserves,  is  the  fact  of  our  seeing 
Erect  vision  objects  erect  when  their  images  on  the  retina  are  inverted.     Erect,  means  nothing  else  than  having    the  head 
by  an  mver-  farther  from  the  ground,  and  the  feet  nearer,  than  any  other  part.     Now,  the  earth,  and  the  objects  which  stand 

mage'  on  it,  preserve  the  same  relative  situation  in  the  picture  on  the  retina  that  they  do  in  nature.  In  that  picture, 
men,  it  is  true,  stand  with  their  heads  downwards  ;  but  then,  at  the  same  time,  heavy  bodies  fall  upwards  ;  and 
the  mind,  or  its  deputy,  the  nerve,  which  is  present  in  every  part  of  the  picture,  judges  only  of  the  relations  of  its 
parts  to  one  another.  How  these  parts  are  related  to  external  objects,  is  known  only  by  experience,  and  judged 
of  at  the  instant  only  by  habit. 

366.  There  is  one  remarkable  fact  which  ought  not  to  escape  mention,  even  in  so  brief  an  abstract  of  the  doctrine 
Punctuin      of  vision  as  the  present.it  is,  that  the  spot  Q,  at  which  the  optic  nerve  enters  the  eye,  is  totally  insensible  to  the 
catcum.        stimulus  of  light,  for  which  reason  it  is  called  the  punctum  ccecum.     The  reason  is  obvious :  at  this   point  the 

nerve  is  not  yet  divided  into  those  almost  infinitely  minute  fibres,  which  are  fine   enough  to  be   either  thrown 

into  tremors,  or  otherwise  changed  in  their  mechanical,  chemical,  or  other  state,  by  a  stimulus  so  delicate  as  the 

Experiment  rays  of  light.     The  effect,  however,  is  curious  and  striking.     On  a  sheet  of  black  paper,  or  other  dark  ground, 

proving  its    place  two  white  wafers,  having  their  centres   three  inches  distant.     Vertically   above   that  to  the  left,  hold  the 

existence.     rfg^f  eye>  at  ^  3  inches  from  it,  and  so  that  when  looking  down   on   it,  the   line  joining  the  two  eyes  shall  be 

parallel  to  that  joining  the  centre  of  the  wafers.     In  this  situation  closing  the  left  eye,  and  looking  full  with  the 

right   at  the  wafer  perpendicularly  below  it,  this   only   is  seen,  the  other  being  completely   invisible.     But  if 

removed  ever  so  little  from  its  place,  either  to  the  right  or  left,  above  or  below,  it  becomes  immediately  visible, 

and  starts,  as  it  were,  into  existence.     The  distances  here  set  down  may  perhaps  vary  slightly  in  different  eyes. 

367.  It  will  cease  to  be  thought  singular,  that  this  fact,  of  the  absolute  invisibility  of  objects   in  a  certain  point   of 
the  field  of  view   of  each  eye,  should  be  one  of  which  not  one  person  in  ten  thousand   is  apprized,  when  we 
learn,  that  it  is  not  extremely  uncommon  to  find  persons  who  have  for   some  time  been   totally  blind  with   one 
eye  without   being  aware  of   the  fact.     One  instance  has  fallen  under  the   knowledge  of  the  writer  of  these 
pages. 

368.  In  the  eyes  of  fishes,  the  humours  being  nearly  of  the  refractive  density  of  the  medium  in  which  they  live,  the 
Eyes  of        refraction  at  the  cornea  is  small,  and  the  work  of  bringing  the  rays  to  a  focus   on  the  retina   is   almost  wholly 

performed  by  the  crystalline.  This  lens,  therefore,  in  fishes  is  almost  spherical,  and  of  small  radius,  in  compa- 
rison with  the  whole  diameter  of  the  eye.  Moreover,  the  destruction  of  spherical  aberration  not  being  producible 
in  this  case  by  mere  refraction  at  the  cornea,  the  crystalline  itself  is  adapted  to  execute  this  necessary  part  of  the 
process,  which  it  does  by  a  very  great  increase  of  density  towards  the  centre.  (Brewster,  Treatise  on 
New  Philosophical  Instruments,  p.  268.)  The  fibrous  and  coated  structure  of  the  crystalline  lens  is  beautifully 
shown  in  the  eye  of  a  fish  coagulated  by  boiling. 

369  The  same  scientific  principles  which  enable  us  to  assist  natural  imperfections  of  sight,  can  be  employed 'in 

giving  additional  power  to  this  sense,  even  in  individuals  who  enjoy  it  naturally  in  the  greatest  perfection.  It 
being  once  understood,  that  the  image  on  the  retina  is  that  which  we  really  see,  it  follows,  that  if  by  any  means 
we  can  render  this  image  brighter,  larger,  more  distinct  than  in  the  natural  state  of  the  organ,  we  shall  see  objects 
brighter  than  in  their  natural  state,  enlarged  in  dimension,  and,  therefore,  capable  of  being  examined  more  in 
detail,  or  more  sharply  defined  and  clearly  outlined.  The  means  which  the  principles  already  detailed  put  in  our 
power,  for  the  accomplishment  of  such  ends,  are  the  concentration  of  more  rays  than  enter  the  natural  eye  by 
lenses ;  the  enlargement  of  the  image  on  the  retina,  by  substituting  for  the  object  seen  an  image  of  it,  either 
larger  than  the  object  itself,  or  capable  of  being  brought  nearer  to  us  ;  and  the  destruction  of  aberration,  by 
properly  adapting  the  figure  and  materials  of  our  instruments  to  the  end  proposed. 

370.  Proposition.     The  apparent  magnitude  of  a  rectilinear  object  is  measured  by  the  angle  subtended  by   it  at 


LIGHT.  401 

Light,     the  centre  of  the  eye,  or  by  the  linear  magnitude  of  its  image  on  the  retina,  and  is  therefore  proportional    Part  * 
—  v—  ^        linear  magnitude  of  object  *"•  "V 

JQ    -  ,  --  —  -  , 

its  distance  from  the  eye 

The  centre  of  the  eye,  in  its  optical  sense,  is  a  point  nearly  in  the  centre  of  the  pupil  in  the  plane  of  the  iris, 
and  the  image  of  any  external  object  P  Q,  being  formed  at  the  bottom  of  the  eye  at  p  q,  by  rays  crossing  there,  Fig.  72. 

p  E 
must  subtend  the  same  angle  ;  so  that  p  q  =  P  Q  .     -. 


Carol.    If  the  object  be  so  distant  that  the  rays  from  each  point  of  it  may  be  regarded  as  parallel,  the  angular      371. 
diameter  of  the  object  is  measured  by  the  inclination  of  rays  of  its  extreme  pencils  to  each  other.     Whenever, 
therefore,  the  eye  sees  by  parallel,  or  very  nearly  parallel,  rays,  the  apparent  magnitude  of  the  object  seen,  is 
measured  by  the  inclination  of  its  extreme  pencils,  and  the  object  itself  is  referred  to  an  infinite  distance,  or  to  the 
concave  surface  of  the  heavens. 

Prop.     When  a  convex  lens  is  placed  between  the  eye  and  any  object,  so  as  to  have  the  object  at  a  distance      372. 
from  the  lens  equal  to  its  focal  length,  it  will  be  distinctly  seen  by  an  eye  capable  of  converging  parallel  rays,  and 
will  appear  enlarged  beyond  its  natural  size. 

Let  P  Q  be  the  object,  C  the  lens,  and  E  the  centre  of  the  eye.  Since  the  object  is  in  the  focus  of  the  lens,  Fig.  73. 
the  rays  of  a  pencil  diverging  from  any  point  P  in  it,  will  emerge,  parallel  to  P  C,  and  to  each  other  ;  they  will, 
therefore,  after  refraction  in  the  eye,  be  brought  to  converge  on  the  retina  to  a  point  p,  such  that  E  p  is  parallel 
to  P  C.  Similarly,  rays  from  Q  will,  after  refraction  through  the  lens  and  eye,  converge  to  q  ;  such  that  E  q  is 
parallel  to  Q  C.  Thus,  a  distinct  image  will  be  formed  atp  q  on  the  retina,  and  the  apparent  angular  magnitude 
of  the  object  seen  through  the  lens  will  be  the  angle  q  E  p.  Now  this  is  equal  to  P  C  Q,  or  the  angle  subtended 
by  the  object  at  the  centre  of  the  lens,  and  is,  therefore,  greater  than  P  E  Q,  or  that  subtended  by  it  at  the  centre 
of  the  eye,  because  the  lens  is  between  the  eye  and  object. 

Hence,  the  nearer  the  eye  is  to  the  lens,  the  less  will  be  the  difference  between  the  apparent  magnitudes  of  the      373. 
object,  as  seen  with  and  without  the  lens  interposed.    But  if  the  lens  be  of  shorter  focus  than  the  least  distance  at 
which  the  eye  can  see  distinctly,  there  will  be  this  essential  difference  between  vision  with  and  without  the  lens, 
that  in  the  former  case  the  object  is  seen  distinctly,  and  well-defined  ;   while  in  the  latter,  or  with  the  naked  eye, 
it  will  be  indistinct  and  confused,  and  the  more  so  the  nearer  it  is  brought. 

Hence,  by  the  use  of  a  convex  lens  of  short  focus,  objects  may  be  seen  distinct,  and  magnified  to  any  extent  we      374. 
please  :  for  let  L  be  the  power,  or  reciprocal  focal  length  of  the  lens,  and  D  the  greatest  proximity  of  the  object  By  a  con- 
to  the  centre  of  the  eye  at  which  it  can  be  seen  distinctly  without  a  lens.     Then  we  shall  have  L  :  D  :  :  angle  vex  lens  of 
p  E  q  :  angle  subtended  by  the  object   at  the  proximity  D  ;    and,  therefore,  :  :  apparent  linear  magnitude  of  s*1?rt  focus 

object  seen  through  the  lens  :  apparent  linear  magnitude  at  proximity  D,  with  the  naked  eye.     Therefore  -fj~is 

the  ratio  of  these  magnitudes,  or,  as  it  is  called,  the  magnifying  power  of  the  lens,  beyond  that  of  the  naked  eye.,  Magnifying 
at  its  greatest  proximity.  power. 

Carol.     D  being  given,  the  magnifying  power  is  as  L,  or  as  (/*  —  1)  (R'  —  R").     This  explains  the  use  of  the      375. 
word  power  in  the  foregoing  sections.     Whatever  we  have  demonstrated  of  the  powers  of  lenses  in  the  foregoing  Magnifying 
pages,  is  true  of  magnifying  powers.     Thus  the  sum    of  the  magnifying  powers  of  two  convex  lenses  is  the  P0  'er  of,  a 
magnifying  power  of  the  two  combined.     If  one  be  concave,  its  magnifying  power  is  to  be  regarded  as  negative,  i^nse™  ° 
and  instead  of  their  sum  we  must  take  their  difference. 

Prop.     To  express,  generally,  the  visual  angle  under  which  a  small  object  placed  at  any  distance  from  a  lens,      376. 
and  seen  by  an  eye  any  how  situated,  appears,  supposing  it  seen  distinctly. 

Let  P  Q,  fig.  74,  75,  76,  77,  be  the  object,  E  the  lens,  O  the  eye,  andp  q  the  image.     Put  =  D,  -  T^'rr'  7°' 

E  Q  E  q       '     • 

1  Visual 

=  />   T-.  ~   =  e;  e  being  reckoned  in  the  same  direction  from  the  centre  of  the  lens  that  D  and/  are.     Then  anele- 
&  O 

the  visual  angle  under  which  the  image  is  seen  is  q  O  p,  and  we  have,  therefore,  visual  angle  (=  A)  =  ?p  = 

_>T,?PT,  —  .  But,  qp  =  Q  P.  —  -^  =  Q  P  .  —f  =  O  .  -f-  putting  O  for  Q  P  the  linear  magnitude  of  the  Vision 
UJi  —  £4  q  tQ  /  /  through 

1  1  f—e  convex 

object  ;  and,  moreover,  O  E  —  E  q  =  —  --  -=-  =—^  —  ,  therefore  we  have,  lenses. 

e          J  J  e 


/    '  /—  e  'L  +  D  —  e 

when  L,  as  all  along,  represents  the  power  of  the  lens.     Now  O  .  D  is  the  visual  angle  of  the  object,  as  seen 

Q  P 

from  the  centre  of  the  lens ;  therefore,  putting  O  .  D,  or  — —  =  (A)  we  get 

Q  E 


VOL.  iv.  3  a 


402 


LIGHT. 


Light. 


377. 

Through 
concave. 

378. 

Inreflectors. 


In  concave  lenses,  the  images  of  distant  objects  are  formed  erect,  and  on  the  same  side  of  the  lens  with  the  object. 
If,  therefore,   such  a  lens  be  held  between  the  eye  and   distant  objects  at  a  sufficient  distance  from  the  eye  for  ' 
distinct  vision,  the  objects  will  be  seen  erect,  and  diminished  in  magnitude.     In  this  case,  eis  positive,  and  L  and 
D  both  negative ;  therefore  L  +  D  —  e  is  a  negative  quantity,  greater  (without  regard  to  the  sign)  than  e,  and, 
consequently,  A  is  negative,  and  less  than  (A). 

In  reflectors,  f  =  2  R  —  D,  and,  therefore, 


(*> 


Part  I. 


In  a  convex  reflector,  e  is  necessarily  negative,  at  least  if  the  mirror  be  made  of  metal,  because  the  eye  must  be 


379. 
General 
principles 
of  tele- 
scopes. 


380. 

Astronomi- 
cal tele- 
•cope. 

Pig.  80, 81 
381. 


Field  of 
view. 


on  the  side  of  the  surface  towards  the  incident  light ;    and,  therefore,  2  R  —  e  is  positive,  and 

2  R  —  D  —  e 

will  be  greater  or  less  than  unity,  according  to  the  value  of  2  R  —  D  —  e.  In  concave  reflectors,  R  is 
negative,  and  e  is  also  negative  for  the  same  reason  as  in  concave  ;  therefore  the  sign  and  magnitude  of 
A  in  this,  as  well  as  the  former  case,  may  vary  indefinitely,  according  to  the  place  of  the  eye,  the  image, 
and  the  object.  The  varieties  of  these  cases  are  represented  in  fig.  78  and  79. 

If  the  image,  instead  of  being  seen  directly  by  the  naked  eye,  be  seen  through  the  medium  of  another 
lens  or  reflector,  so  placed  as  to  cause  the  pencils  diverging  primarily  from  each  point  of  the  object,  to 
emerge  finally,  either  exactly  parallel,  or  within  such  limits  of  convergence  or  divergence  as  the  eye  can 
accommodate  itself  to,  the  object  will  be  seen  distinctly,  and  either  larger  or  smaller  than  it  would  be  seen  by  the 
unassisted  eye,  according  to  the  magnitude  of  the  image,  and  the  power  of  the  lens  or  reflector  used  to  view 
it.  This  is  the  principle  of  all  telescopes  and  microscopes.  As  most  eyes  can  see  with  parallel  rays,  they  are 
so  constructed  as  to  make  parallel  pencils  emerge  parallel  ;  and  a  mechanical  adjustment  allows  such  a  quantity 
of  motion  of  the  lenses  or  reflectors  with  respect  to  each  other,  as  to  give  the  rays  a  sufficient  degree  of  conver- 
gence or  divergence  as  may  be  required. 

In  the  common  refracting,  or,  as  it  is  sometimes  called,  the  astronomical  telescope,  the  image  is  first 
formed  by  a  convex  lens,  and  is  viewed  through  a  convex  lens,  placed  at  a  distance  from  the  other  nearly 
equal  to  the  sum  of  their  focal  lengths.  The  lens  which  forms  the  image  is  called  the  object-glass,  and  that 
through  which  it  is  viewed,  the  eye-glass  of  the  telescope.  If  the  latter  be  concave,  the  telescope  is  said  to 
be  of  the  Galilean  construction,  such  having  been  the  original  arrangement  of  Galiltco's  instruments.  The 
situation  of  the  lenses,  and  the  course  of  the  rays  in  these  two  constructions,  are  represented  in  fig.  80  and  81. 

In  the  former  construction,  let  P  Q  be  the  object.  Draw  Q  O  G  through  the  centres  of  the  object  and 
eye-glass,  and  this  line  will  be  the  axis  of  the  telescope.  From  R  any  point  in  the  object  draw  P  O  p  through 
the  centre  O  of  the  object-glass,  and  meeting  p  q,  a  line  through  q,  the  focus  of  the  point  Q,  perpendicular 
to  the  axis  in  p,  then  will  p  q  be  the  image  of  P  Q.  Let  P  A,  P  B  be  the  extreme  rays  of  the  pencil  diverging 
from  P,  and  incident  on  the  object-glass,  and  they  will  be  refracted  to  and  cross  at  p.  Hence,  unless 
the  diameter  of  the  eye-glass  b  G  a  be  such,  that  the  ray  A.p  a  shall  be  received  on  it,  the  point  p  will  be 
seen  less  illuminated  than  the  point  Q  in  the  centre  of  the  object,  and  if  it  be  so  small  that  the  line 
Bp  produced  does  not  meet  it,  then  none  of  the  rays  from  P  can  reach  the  eye  at  all.  Thus,  the  field  of 
view,  or  angular  dimensions  of  the  object  seen,  is  limited  by  the  aperture  of  the  eye-glass.  To  find  its  extent, 
then,  join  B  b,  A  a,  opposite  extremities  of  the  object  and  eye-glass,  meeting  the  image  in  r  and  p,  and  the 
axis  in  X,  then  r  p  is  the  whole  extent  of  the  image  which  is  seen  at  all,  and  the  angle  p  O  r,  which  is 
equal  to  P  O  R,  is  the  angular  extent  of  the  field  of  view.  Now  we  have  AB:a6::OX:GX,  and, 


therefore,  AB+a5:AB::OG:OX,  whence  we  get  O  X  = 


AB 


A  B  +  ab 


.  OG;  GX  = 


a  b 


AB  +  ab 


O  G.     But  we  have,  moreover,  X  q  =  O  q  —  O\;  pr  =  ab.          ,  and  angle  r  O  p  =         .      To    express 

*  i    A.  \J  O 

this  algebraically,  put 

Diameter  of  object-glass  =  a ;  Power  of  object-glass  =  L 
Diameter  of  eye-glass  =  ft  ;       Power  of  eye-glass  =  I. 


Then 


IL  +  l 

This  last  is  the  linear  magnitude  of  the  visible  portion  of   the  image ;    and  it  is,  as  we  see,  symmetrical 
both  with  respect  to  the  eye-glass  and  object-glass. 

382.  Now  from  this  it  is  easy  to  deduce  both  the  field  of  view  and  magnifying  power  of  the  telescope  ;    for  the 

former  is  equal  to  the  angle  subtended  by  p  r,  at  the  centre  of  the  object-glass,  and  the  latter  is  obtained  from 
the  former,  when  the  angle  rGp  subtended  at  the  centre  of  the  eye-glass  is  obtained.     But  we  have 


LIGHT.  403 


F 

fw\  formulas 

rGp  I          '  for  field  of 

therefore  magnifying  power  =  _^-  =  —J  ««w  ^ 

power. 

Hence  we  see,  that  the  greater  the  power  of  the  eye-glass  is,  compared  with  that  of  the  object-glass,  the  greater 
the  magnifying  power  of  the  telescope  ;  or,  in  other  words,  the  greater  the  focal  length  of  the  object  glass  com- 
pared with  that  of  the  eye-glass. 

The  pencils  of  rays  after  refraction  at  the  eye-glass  will   emerge  parallel,  and  therefore  proper  for  distinct      383. 
vision  to  an  eye  properly  placed  to  receive  them.     Now  the  eye  will  receive  both  the  extreme  rays  6  R'  and  a  V  Distance 
of  the  pencils  diverging  from  r  and  p,  if  it  be  placed  at  their  point  of  concourse  E  ;   but  since  6  E  is  parallel  to  of  fiye- 
r  G,  and  a  E  to  p  G,  we  have 


„ 


If  the  eye  be  placed  nearer  to,  or  farther  off  from,  the  eye-glass  than  this  distance,  it  will   not  receive  the      384. 
extreme  rays,  and  thejield  of  view,  or  visible  area  of  the  object,  will  be  lessened.     In  the  construction  of  convex 
single  eye-pieces,  therefore,  care  must  be  taken  to  prolong  the  tube  which  carries  them,  (as  in  the  figure,)  so  that 
when  the  eye  is  applied  close  to  its  end,  it  shall  still  be  at  this  precise  distance  from  the  glass. 

If  the  telescope  be  inverted,  and  the  eye  applied  behind  the  object-glass,  it  is  evident  that  it  will  remain  a      385. 

T  Inversion  of 

telescope,  but  its  magnifying  power  will  be  changed  to  -  -  ;  so  that,  if  it  magnified  before,  it  will  diminish  objects  telesc°Pes- 

p 

now,  and  the  field  of  view  will  be  proportionally  increased.  In  this  way,  beautiful  miniature  pictures  of  distant 
objects  may  be  seen. 

If  the  telescope,  instead  of  being  turned  on  objects  so  distant  as  that  the  pencils  flowing  from  them  may  be      386. 
regarded  as  parallel,  be  directed   to  near   objects,  the  distance   between  the  object-glass  and  eye-glass  must  be  Adjust- 
lengthened  so  as  to  bring  the  image  exactly  into  the  focus  of  the  latter.     To  accomplish  this,  the   eye-glass  is  mi 
generally-  set  in  a  sliding  tube  movable  by  a  rack-work,  or  by  hand.     The  same  mechanism  serves  also  to  adjust 
the  telescope  for  long  or  short-sighted  persons.     The  former  require  parallel  or  slightly  divergent  rays,  the  latter 
very  divergent  ;  and  to  obtain  the  necessary  divergence  for  the  latter,  the  eye-glass  must  be  brought  nearer  the 
object-glass. 

The  same  theory  and  formulae  apply  to  the  second,  or  Galilaean,  construction,  only  recollecting  that  in  this  case  L,      387. 
the  power  of  the  eye-glass,  is  negative.     In  this  case,  therefore,  the  value  of  G  E  is  negative,  or  the  eye  should  J^*" 
be  placed  between  the  object-glass  and  eye-glass  ;  but,  as  that  is  incompatible  with  the  other  conditions,  in  order 
to  get  as  great  a  field  of  view  as  possible,  the  eye  must  be  brought  as  near  to  its  proper  place  as  possible,  and 
therefore  close  to  the  eye-glass. 

In  the  astronomical  telescope  objects  are  seen  inverted,  in  the  Galilaean,  erect  ;  for,  in  the  former,  the  rays      388. 
from  the  extremities  of  the  object  have  crossed  before  entering  the  eye,  in  the  latter,  not. 

If  the  object  be  brought  nearer  the  object-glass,  the  magnifying  power  is  increased  ;    because  in  this  case      339. 

•  Micro- 

(calling  D  the  proximity  of  the  object)   -  -  —   expresses  the  magnifying  power,  as  is  easily  seen  from  what  has  scopes. 

J_j  —  D 

been  said  Art.  382.  Thus  a  telescope  used  for  viewing  very  near  objects  becomes  a  microscope.  The  ordinary 
construction  of  the  compound  microscope  is  nothing  more  than  that  of  the  astronomical  telescope  modified  for 
the  use  it  is  intended  for.  The  object-glass  has  in  this  instrument  a  much  greater  power  than  the  eye-glass,  so 
that,  when  employed  for  viewing  distant  objects,  it  acts  as  a  telescope  inverted,  and  requires  to  be  greatly 

shortened.  But  for  near  objects,  as  D  increases,  I  —  D  diminishes,  and  the  fraction  _  —  may  be  increased 
to  any  amount,  by  bringing  the  object  nearer  to  the  object-glass,  and  at  the  same  time  lengthening  the  interval 

between   the   lenses,  which   is    expressed   by  -  +    -  .      But  as  this   requires  two   operations,  it  is 

\j  —  iJ  I 

usual  to  leave  the  latter  distance  unaltered,  and  vary,  by  a  screw  or  rack-work,  only  the  former.  Fig.  82  is  a  Fig.  82. 
section  of  such  an  instrument.  It  is,  however,  convenient  to  have  the  power  of  lengthening  and  shortening  the 
distance  between  the  glasses,  as  by  this  means  any  magnifying  power  between  the  limits  corresponding  to  the 
extreme  distances  may  be  obtained  ;  and  if  a  series  of  object-glasses  be  so  selected,  that  the  greatest  power 
attainable  by  one  within  the  limits  of  the  adjustment  in  question,  shall  just  surpass  the  least  obtainable  by  the 
next,  and  so  on,  we  may  command  any  power  we  please.  Such  a  series  is  usually  comprised  in  a  small  revolving 
plate  containing  cells,  each  of  which  can  be  brought  in  succession  into  the  axis  of  the  microscope  by  a  simple 
mechanism. 

In  the  reflecting  telescope,  of  the  most  simple  construction,  the  image  is  formed  by  a  concave  mirror,  and      390 
viewed  by  a  convex  or  concave  "eye-glass,  as  in  refracting  telescopes  ;  but  since  the  head  of  the  observer  would  Reflecting 
intercept  the  whole  of  the  incident  light  in  small  telescopes,  and  a  great  part  of  it  in  large  ones,  the  axis  of  the  telescope. 
reflector  itself  is  turned  a  little  obliquely,  so  as  to  throw  the  image  aside,  by  which  it  can  be  viewed  with  little  or 
no  loss  of  light.     The  inconvenience  of  this  is  a  little  distortion  of  the  image,  caused  by  the  obliquity  of  the  rays  ; 

3  o  2 


404 


LIGHT. 


Herschelian 
construc- 
tion. 

391. 


Light  but  as  such  telescopes  are  only  used  of  a  great  size,  and  for  the  purpose  of  viewing  very  faint  celestial  objects, 
•-^•sr-—''  in  which  the  light  diffused  by  aberration  is  insensible,  little  or  no  inconvenience  is  found  to  arise  from  this  cause. 
Simplest,  or  Such  is  the  construction  of  the  telescopes  used  by  Sir  William  Herschel  in  his  sweeps  of  the  heavens. 

To  obviate  the  inconvenience  of  the  stoppage  of  rays  by  the  head,  Newton,  the  inventor  of  reflecting  tele- 
scopes, employed  a  small  mirror,  placed  obliquely,  as  in  fig.  83,  opposite  the  centre  of  the  large  one.  Thus 
parallel  rays  PA,  PB,  emanating  from  a  point  in  the  axis  of  the  telescope,  are  received,  before  their  meeting,  on 
Newtonian  a  plane  mirror  C  D  inclined  at  45°  to  the  axis,  and  thence  reflected  through  a  tube  projecting  from  the  side  of 
construe-  the  telescope  to  the  lens  G,  and  by  it  refracted  to  the  eye  E.  It  is  manifest,  that  if  the  image  formed  by  the 
mirror  A  B  behind  C  D  be  regarded  as  an  object,  an  image  equal  and  similar  to  it  (Art.  335)  will  be  formed 
at  F,  at  an  equal  distance  from  the  plane  mirror  ;  and  this  image  will  be  seen  through  the  glass  G,  just  as  if  it 
were  formed  by  an  object-glass  of  the  same  focal  length  placed  in  the  prolongation  of  the  axis  of  the  eye-tube, 
beyond  the  small  mirror,  (supposed  away.)  Hence  the  same  propositions  and  formulae  will  hold  good  in  the 
Newtonian  telescope,  as  in  the  astronomical  and  Galilaean,  for  the  magnifying  power,  field  of  view,  and  position 
of  the  eye,  substituting  only  2  R  for  L,  and  2  R  —  D  for  L  —  D,  and  recollecting  that  R  is  negative,  as  the 
mirror  has  its  concavity  turned  towards  the  light. 

392.          The  Gregorian  telescope,  instead  of  a  small  plain  mirror  turned  obliquely,  has  a  small  convex  mirror  with  its 

Gregorian     concavity  turned  towards  that  of  the  large  one,  as  in  fig.  84  ;  but  instead  of  being  placed  at  a  distance  from  the 

telescope,     large  one  equal  to  the  sum  of  the  focal  lengths,  the  distance  is  somewhat  greater  ;  hence  the  image  p  q,  formed 

Fig.  84.       jn  the  focus  of  the  great  mirror,  being  at  a  distance  from  the  vertex  of  the  small  one  greater  than  its  focal  length, 

another  image  is  formed  at  a  distance,  viz.  at  or  near  the  surface  of  the  great  mirror,  at  r  s.     In  the  centre  of  the 

large  mirror  there  is  a  hole  which  lets  pass  the  rays  to  an  eye-lens  g.     The  adjustment  to  parallel  or  diverging 

rays,  or  for  imperfect  eyes,  is  performed  by  an  alteration  of  the  distance  between  the  mirrors  made  by  a  screw. 

363.          The  Cassegrainian  construction  differs  in  no  respect  from  the  Gregorian,  except  that  the  small  mirror  is  convex 

Cassegra'm-  and  receives  the  rays  before  their  convergence  to  form  an  image.     The  magnitude  of  the  field,  the  distance  of  the 

'*"•  eye  and  of  the  mirrors  from  each  other,  are  easily  expressed  in  these  constructions  ;  the  latter  being  derived  from 

the  former  by  a  mere  change  of  sign  in  the  curvature  of  the  small  mirror.     Let  then  R'  and  R"  be  the  curvatures 

of  the  two  mirrors,  then  in  the  Gregorian  telescope  R'  is  negative  and  R"  positive  ;  and  if  we  put  t  for  the 

distance  between  their  "surfaces,  (t  being  negative,  because  the  second  reflecting  surface  lies  towards  the  incident 

light)  we  shall  have  for  an  object  whose  proximity  is  D 


Part  I. 


D'=D; 


=  2R'-D  =2R'-D;        /"=2R"—  D"; 


adopting  the  formulae  and  notation  of  Art.  251.     Now  these  give,  by  substitution, 

2  R'  -  D  2  R'  -  D 


D"  = 


1  -  <(2R'-  D)    ' 
2  R"  -  2  R' 


/"_2R"         l_t(2R,_0) 
D  —  2  t  (2  R'  —  D)  .  R" 


1 


394. 


<(2R'  — D) 

This  is  the  reciprocal  distance  of  the  second  image  from  the  second  reflecting  surface.     If  we  wish  that  the  image 
to  be  viewed  by  the  eye-lens  should  fall  just  on  the  surface  of  the  large  mirror,  we  have  only  to  put  f"  = 

(because  f"  is  positive,  and  t  negative.)     For  parallel  rays  this  gives 

R'  R"  .  t*  +  (4  R'  -  2  R")  t  -  I  =  o;  (g) 

whence  t  may  be  found  when  R'  and  R"  are  given,  or  vice  versd. 

The  description  of  other  optical  instruments,  and  of  the  more  refined  construction  of  telescopes,  &c.  must  be 
deferred  till  we  are  farther  advanced  in  our  account  of  the  physical  properties  of  light,  and  especially  of  the 
different  refrangibility  of  its  rays  and  their  colours,  which  will  form  the  object  of  the  next  part. 


Light 


LIGHT.  405 

Part  II. 


PART  II. 
CHROMATICS. 

§  I.     Of  the  Dispersion  of  Light. 

HITHERTO  we  have  regarded  the  refractive  index  of  a  medium  as  a  quantity  absolutely  given  and  the  same  for      395 
all  rays  refracted  by  the  medium.     In  nature,  however,  the  case  is  otherwise.     When  a  ray  of  light  falls  obliquely  General 
on  the  surface  of  a  refracting  medium,  it  is  not  refracted  entirely  in  one  direction,  but  undergoes  a  separation  phenome- 
into  several  rays,  and  is  dispersed  over  an  angle  more  or  less  considerable,  according  to  the  nature  of  the  medium  °°t°o°  ^p*" 
and  the  obliquity  of  incidence.     Thus   if  a   sunbeam   S  C  be  incident  on  the  refracting  surface  A  B,  and  be  ray  jnto 
afterwards  received  on  a  screen  R  V,  (fig.  85,)  it  will,  instead,  of  a  single  point  on  the  screen  as  R,  illuminate  colours. 
a  space  R  V  of  a  greater  extent  the  greater  is  the  angle  of  incidence.     The  ray  S  C,  then,  which,  before  refraction  Fig-  85. 
was  single,  is  separated  into  an  infinite  number  of  rays  C  R,   CO,    C  Y,  &c.  each  of  which  is  refracted 
differently  from  all  the  rest. 

The  several  rays  of  which  the  dispersed  beam  consists,  are  found  to  differ  essentially  from  each  other,  and  from  396. 
the  incident  beam,  in  a  most  important  physical  character.  They  are  of  different  colours.  The  light  of  the  sun 
is  white.  If  a  sunbeam  be  received  directly  on  a  piece  of  paper,  it  makes  on  it  a  white  spot ;  but  if  a  piece  of 
white  paper  (that  is,  such  as  by  ordinary  daylight  appears  white)  be  held  in  the  dispersed  beam,  as  R  V,  the 
illuminated  portion  will  be  seen  to  be  differently  coloured  in  different  parts,  according  to  a  regular  succession  of 
tints,  which  is  always  the  same,  whatever  be  the  refracting  medium  employed. 

To  make  the  experiment  in  the  most  striking  and  satisfactory  manner,  procure  a  triangular  prism  of  good  397. 
flint-glass,  and  having  darkened  a  room,  admit  a  sunbeam  through  a  small  round  hole  O  P  in  the  window  FlS-  8S 
shutter.  If  this  be  received  on  a  white  screen  D  at  a  distance,  there  will  be  formed  a  round  white  spot,  or 
image  of  the  sun,  which  will  be  larger  as  the  paper  is  farther  removed.  New  in  the  beam  before  the  screen 
place  the  prism  ABC,  having  one  of  its  angles  C  downwards  and  parallel  to  the  horizon,  and  at  right  angles 
to  the  direction  of  the  sunbeam,  and  let  the  beam  fall  on  one  of  its  sides  B  C  obliquely.  It  will  be  refracted 
and  turned  out  of  its  course,  and  thrown  upwards,  pursuing  the  course  FOR,  and  may  be  received  on  a  screen 
E  properly  placed.  But  on  this  screen  there  will  no  longer  be  seen  a  white  round  spot,  but  a  long  streak,  or, 
as  it  is  called  in  Optics,  a  spectrum  R  V  of  most  vivid  colours,  (provided  the  admitted  sunbeam  be  not  too  large, 
and  the  distance  of  the  screen  from  the  prism  considerable.)  The  tint  of  the  lower  or  least  refracted  extremity 
R  is  a  brilliant  red,  more  full  and  vivid  than  can  be  produced  by  any  other  means,  or  than  the  colour  of  any 
natural  substance.  This  dies  away  first  into  an  orange,  and  this  passes  by  imperceptible  gradations  into  a  fine 
pale  straw-yellow,  which  is  quickly  succeeded  by  a  pure  and  very  intense  green,  which  again  passes  into  a  blue, 
at  first  of  less  purity,  being  mixed  with  green,  but  afterwards,  as  we  trace  it  upwards,  deepening  to  the  purest 
indigo.  Meanwhile,  the  intensity  of  the  illumination  is  diminishing,  and  in  the  upper  portion  of  the  indigo  tint 
is  very  feeble,  but  it  is  continued  still  beyond,  and  the  blue  acquires  a  pallid  cast  of  purplish  red,  a  livid  hue 
more  easily  seen  than  described,  and  which,  though  not  to  be  exactly  matched  by  any  natural  colour,  approaches 
most  nearly  to  that  of  a  fading  violet :  "  tinctus  mold  pallor." 

If  the  screen  on  which  the  spectrum  be  received  have  a  small  hole  in  it,  too  small  to  allow  the  whole  of  the      398. 
spectrum  to  pass,  but  only  a  very  narrow  portion  of  it,  as  X,  (fig.  87,)  the  portion  of  the  beam  which  goes  to  Insulation 
form  that  particular  spot  X  may  be  received  on  another  screen  at  any  distance  behind  it,  and  will  there  form  a  ™  each 
spot  d  of  the  very  same  colour  as  the  part  X  of  the  spectrum.     Thus  if  X  be  placed  in  the  red  part  of  the  co 
spectrum  the  spot  d  will  be  red  ;  if  in  the  green,  green ;    and  in  the  blue,  blue.     If  the  eye  be  placed  at  d,  it 
will  see  through  the  hole  an  image  of  the  sun  of  dazzling  brightness ;  not,  as  usually,  white,  but  of  the  colour 
which  goes  to  form  the  spot  X  of  the  spectrum.     Thus  we  see,  that  the  joint  action  of  all   the  rays   is  not 
essential  to  the  production  of  the  coloured  appearance  of  the  spectrum,  but  that  one  colour  may  be  insulated 
from  the  rest,  and  examined  separately. 

If,  instead  of  receiving  the  ray  X  d,  transmitted  through  the  hole  X,  on  a  screen  immediately  behind  it,  it  be      399. 
intercepted  by  another  prism  a  c  b,  it  will  be  refracted,  and  bent  from  its  course,  as  in  ~X.fgx  ;  and  after  this  Second 
second  refraction  may  be  received  on  a  screen  e.     But  it  is   now  observed  to  be  no  longer  separated  into  a  Iefraction 
coloured  spectrum  like  the  original  one  R  V,  of  which  it  formed  a  part.     A  single  spot  x  only  is  seen  on  the  5™^"^,"^ 
screen,  the  colour  of  which  is  uniform,  and  precisely  that  which  thn  part  X  of  the  spectrum  would  have  had,  change  of 
were  it  intercepted  on  the  first  screen.     It  appears,  then,  that  the  ray  which  goes  to  form  any  single  point  of  the  colour, 
spectrum  is  not  only  independent  of  all   the  rest,  but  having  been  once  insulated  from  them,  is  no  longer 
capable  of  further  separation  into  different  colours  by  a  second  refraction. 

This  simple,  but  instructive  experiment,  then,  makes  us  acquainted  with  the  following  properties  of  light : 


406  LIGHT. 

Light.          1.  A  beam  of  white  light  consists  of  a  great  and  almost  infinite  variety  of  rays  differing  from  each  other  in     P»rt  II. 
>— — V— '  colour  and  refrangibility.  V—""V~~' 

For  the  ray  S  F  from  any  one  point  of  the  sun's  disc,  which   if  received  immediately  on   the   screen  would 

m  refraiTJi-  nave  occupied  only  a  single  point  on  it,  or  (supposing  the  hole  in  the  screen  to  have  a  sensible  diameter)  only  a 

bility.       '    space  equal  to   its  area,  is  dilated  into  a  line  V  R  of  considerable  length,   every  point   of  which   (speaking 

loosely)  is  illuminated.     Now   the   rays  which   go  to  V  must  necessarily  have  been  more  refracted  than  those 

which  go  to  R,  which  can  only  have  been  in  virtue  of  a  peculiar  quality  in  the  rays  themselves,  since  the 

refracting  medium  is  the  same  for  all. 

401.  2.  White  light  may  be  decomposed,  analyzed,  or  separated  into  its  elementary  coloured  rays  by  refraction.     The 
act  of  such  separation  is  called  the  dispersion  of  the  coloured  rays. 

402.  3.  Each  elementary  ray  once  separated  and  insulated  from  the  rest,  is  incapable  of  further  decomposition  or 
analysis  by  the  same  means.     For  we  may  place  a  third,  and  a  fourth,  prism  in  the  way  of  the  twice  refracted 
ray  g  x,   and  refract  it  in  any  way,  or  in   any  plane  ;    it  remains  undispersed,  and  preserves  its  colour  quite 
unaltered. 

403.  4.  The  dispersion  of  the  coloured  rays  takes  place  in  the  plane  of  the  refraction ;  for  it  is  found  that  the 
spectrum  V  R  is  always  elongated  in  this  plane.     Its  breadth  is  found,  on  the  other  hand,  by  measurement,  to  be 
precisely  the  same  as  that  of  the  white  image  D,  (fig.   86,)  of  the  sun,  received  on  a  screen   at  a  distance  O  D 
from  the  hole,  equal  to  O  F  +  F  G  4-  G  R,  the  whole  course  of  the  refracted  light,  which  shows  that  the  beam 
has  undergone  no   contraction  or  dilation  by  the  effect  of  refraction  in  a  plane  perpendicular  to  the  plane  of 
refraction. 

404.  To   explain  all  the  phenomena  of  the  colours  produced  by  prismatic  dispersion,  or  of  the  prismatic  colours, 
Index  of      as  they  are  called,  we  need  only  suppose,  with  Newton,  that  each  particular  ray  of  light,  in  undergoing  refraction 
refraction      at  the  surface  of  a  given  medium,  has  the  sine  of  its  angle  of  incidence  to  that  of  refraction  in  a  constant  ratio, 
regarded  as  so  ]ong.  as  the  medium  and  the  ray  are  the  same  ;  but  that  this  ratio  varies  not  only,  as  we  have  hitherto  all  along 

e'  assumed,  with  the  nature  of  the  medium,  but  also  with  that  of  the  ray.  In  other  words,  that  there  are  as  many 
distinct  species,  or  at  least  varieties  of  light,  as  there  are  distinct  illuminated  points  in  the  spectrum  into  which 
a  single  ray  of  white  light  is  dispersed.  This  amounts  to  regarding  the  quantity  fi,  for  any  medium,  not  as  one 
and  invariable,  but  as  susceptible  of  all  degrees  of  magnitude  between  certain  limits  :  one,  the  least  of  which, 
corresponds  to  the  extreme,  or  least  refracted  red  ray  ;  the  other,  the  greatest  value  of  ft,  to  the  extreme  or 
most  refracted  violet.  Each  of  these  varieties  separately  conforms  to  the  laws  of  reflexion  and  refraction  we 
have  already  laid  down.  As  in  Geometry  we  may  regard  a  whole  family  of  curves  as  comprehended  under  one 
equation,  by  the  variation  of  a  constant  parameter  ;  so  in  Optics  we  may  include  under  one  analysis  all  the 
doctrine  of  the  reflexions,  refractions,  and  other  modifications  of  a  ray  of  white  or  compound  light,  by  regarding 
the  refractive  index  /a,  as  a  variable  parameter. 

405  To  apply  this,  for  instance,  to  the  experiment  of  the  prism  just  related :    A  single  ray  of  white  light  being 

supposed  incident  on  the  first  surface,  must  be  regarded  as  consisting  of  an  infinite  number  of  coincident  rays, 
of  all  possible  degrees  of  refrangibility  between  certain  limits,  any  one  of  which  may  be  indifferently  expressed 
by  the  refractive  index  ft.  Supposing  the  prism  placed  so  as  to  receive  the  incident  ray  perpendicularly  on  one 
surface,  then  the  deviation  will  be  given  by  the  equation 

ft .  sin  I  =  sin  (I  +  D) 

I  being  the  refracting  angle  of  the  prism.  D  therefore  is  a  function  of  ft,  and  if  ft  vary  by  the  infinitely  small 
increment  S  ft,  i.  e.  if  we  pass  from  any  one  ray  in  the  spectrum  to  the  consecutive  ray,  D  will  vary  by  6  D, 
and  the  relation  between  these  simultaneous  changes  will  be  given  by  the  equation  resulting  from  the  differen- 
tiation of  the  above  with  the  characteristic  S  :  thus  we  get 

B  ft.  sin  I  =  3D.  cos  (I  +  D)  ;         8  D  =  S  ft  .  —j^——.  (a) 

it  is  evident,  then,  that  as  ft  varies,  D  also  varies ;  and,  therefore,  that  no  two  of  the  refracted  and  coloured  rays 
will  coincide,  but  will  be  spread  over  an  angle,  in  the  plane  of  refraction,  the  greater,  the  greater  is  the  total 
variation  of  ft  from  one  extreme  to  the  other. 

406.  In  order  to  justify  the  term  analysis,  or  decomposition,  as  applied  to  the  separation  of  a  beam  of  white  light 

Analysis       ;nto  coloured  rays,  we  must  show  by  experiment  that  white  light  may  be  again  produced  by  the  synthesis  of  these 

sU^oFwhit"  e'ementarv  rays-     The  experiment  is  easy.     Take  two  prisms  A  B  C,  a  b  c  of  the  same  medium,  and  having 

light  equal  refracting  angles,  and  lay  them  very  near  together,  having  their  edges  turned  opposite   ways,  as  in  fig.  87. 

Fig.  87.       With  this  disposition,  a  parallel  beam  of  white  light  intromitted  into  the  face  A  C  of  the  first  prism,  will  emerge 

from  the  face  6  c  of  the  last,  undeviated,  and  colourless,  as  if  no  prisms  were  in  the  way.     Now  the  dispersion 

having  been  fully  completed  by  the  prism  ABC,  the  rays  in  passing  through  the  thin  lamina  of  air  B  C  a  c  must 

have  existed  in  their  coloured  and  independent  state,  and  been  dispersed  in  their  directions  ;  but  being  refracted 

by  the  second  prism  so  as  to  emerge  parallel,  the  colour  is  destroyed  by  the  mixture  and  confusion  of  the  rays. 

Fig.  88.       To  see  more  clearly  how  this  takes  place  in  fig.  88,  let  S  R  and  S  V  be  two  parallel  white  rays,  incident  on  the 

first  prism,  and  separated  by  refraction  ;  the  former  into  the  coloured  pencil  R  c  v,  the  latter  into  a  pencil  exactly 

similar  to  V  r  c.     Let  Re  be  the  least  refracted  ray  of  the   former  pencil,   and  Vc  the  most  refracted   of  the 

latter.     These,  of  course,  must  meet ;    let  them  meet  in  c,   and  precisely  at  c  apply  the  vertex  of  the  second 

prism,  having  its  side  c  a  parallel  to  C  B,  but  its  edge  turned  in  the  opposite  direction  ;  then  will  the  rays  R  C 

and  V  c,  each  for  itself,  and  independent  of  the  other,  be  refracted  so  as  to  emerge  parallel  to  its  original  direction 


LIGHT.  407 

Light.  S  R,  S  V,  and  the  emergent  rays  will  therefore  be  coincident  and  superimposed  on  each  other  as  cs.  Thus  the  Part  II. 
•••v"»p/  emergent  ray  cs  will  contain  an  extreme  red  and  an  extreme  violet  ray.  But  it  will  also  contain  every  inter-  ^*~-^~ — 
mediate  variety;  for  draw  cf  anywhere  between  cR  and  cV.  Then,  since  the  angle  which  cf  makes  with  the 
surface  B  C  is  greater  than  that  made  by  the  extreme  violet  ray  C  B,  but  less  than  that  made  by  the  extreme 
red,  there  must  exist  some  value  of  fi  intermediate  between  its  extreme  values,  which  will  give  a  deviation  equal 
to  the  angle  between  cf  and  S  Y  parallel  to  S  R.  Consequently,  if  S  Y  be  a  white  ray,  separated  into  the 
pencil  Y  v'  r  by  refraction,  the  coloured  ray  Y/c  of  that  particular  refrangibility  will  fall  on  c,  and  be  refracted 
along  cs.  Every  point  then  of  the  surface  gfh  will  send  to  c  a  ray  of  different  refrangibility,  comprehending  all 
the  values  of  /i  from  the  greatest  to  the  least.  Thus  alt  the  coloured  elements,  though  all  belonging  originally 
to  different  white  rays,  will,  after  the  second  refraction,  coincide  in  the  ray  cs,  and  experience  proves  that  so 
reunited  they  form  white  light.  White  light,  then,  is  re-composed  when  all  the  coloured  elements,  even  though 
originally  belonging  to  separate  white  rays,  are  united  in  place  and  direction. 

In  the  reflexion  of  light,  regarded  as  a  case  of  refraction,  /i  has  a  specific  numerical  value,  and  cannot  vary      407. 
without  subverting  the  fundamental  law  of  reflexion.     Thus,  there  is  no  dispersion  into  colours  produced  by 
reflexion,  because  all  the  coloured  rays  after  reflexion  pursue  one  and  the  same  course.     There  is  one  exception 
to  this,  more  apparent  than  real,  when  light  is  reflected  from  the  base   of  a  prism  internally,   of  which  more 
hereafter. 

The  recomposition  of  white  from  coloured  light  may  be  otherwise  shown,  by  passing  a  small  circular  beam  of      408. 
solar  light  through  a  prism  ABC,  (fig.  89,)  and  receiving  the  dispersed  beam  on  a  lens  E  D  at  some  distance.  Synthesis 
If  a  white  screen  be  held  behind  the  lens,  and  removed  to  a  proper  distance,  the  whole   spectrum  will  be  °<  white 
reunited  in  a  spot  of  white  light.     The  way  in  which  this  happens  will  be  evident  by  considering  the  figure,  in  !'gllt  ^  ' 
which  TE  and  TD  represent  the  parallel  pencils  of  rays  of  any  two  colours  (red  and  violet,  for  instance)  into  'en*' 
which  the  incident  white  beam  S  T  is  dispersed.     These  will  be  collected  after  refraction,  each  in  its  own  proper 
focus  ;  the  former  at  F,  the  latter  at  G  ;  after  which  each  pencil  diverges  again,  the  former  in  the  cone  F  H,  the 
latter  in  G  H.     If  the  screen  then  be  held  at  H,  each  of  these  pencils  will  paint  on  it  a  circle  of  its  own  colour, 
and  so  of  course  will  all  the  intermediate  ones ;  but  these  circles  all  coinciding,  the  circle  H  will  contain  all  the 
rays  of  the  spectrum  confounded  together,  and  it  is  found  (with  the  exception  of  a  trifling  coloured  fringe  about 
the  edges,  arising  from  a  slight  overlapping  of  the  several  coloured  images)  to  be  of  a  pure  whiteness. 

That  the  reunion  of  all  the  coloured  rays  is  necessary  to  produce  whiteness,  may  be  shown  by  intercepting  a      409. 
portion  of  the  spectrum  before  it  falls  on  the  lens.     Thus,  if  the  violet  be  intercepted,  the  white  will  acquire  a  A"  tlle 
tinge  of  yellow;   if  the  blue  and  green  be  successively  stopped,  this  yellow  tinge  will  grow  more  and  more  ruddy,  {Jf 
and  pass  through   orange  to  scarlet  and  blood  red.     If,  on   the  other  hand,  the  red  end  of  the  spectrum  be  wht 
stopped,  and  more  and  more  of  the  less  refrangible  portion  thus  successively  abstracted  from  the  beam,  the  white 
will  pass  first  into   pale  and   then    to  vivid  green,  blue-green,  blue,  and  finally   into  violet.      If   the    middle 
portion  of  the  spectrum  be  intercepted,  the  remaining  rays,  concentrated,  yroduce  various  shades  of  purple,  Al1 
crimson,  or  plum-colour,  according  to  the  portion  by  which  it  is  thus  rendered  deficient  from  white  light ;    and  co]l?u?  Iml' 
by  varying  the  intercepted  rays,  any  variety  of  colours  may  be  produced  ;   nor  is   there  any  shade  of  colour  in  combina- 
nature  which  may  not  thus  be  exactly  imitated,  with  a  brilliancy  and  richness  surpassing  that  of  any  artificial  tions  of  the 
colouring.  prismatic. 

Now,  if  we  consider  that  all  these  shades  are  produced  on  white  paper,  which  receives  and  reflects  to  our  eyes 
whatever  light  happens  to  fall  on  it ;  and  that  the  same  paper  placed  successively  in  the  red,  green,  and  blue 
portion  of  the  spectrum,  will  appear  indifferently  red,  or  green,  or  blue,  we  are  naturally  enough  led  to  conclude, 
that 

The  colours  of  natural  bodies  are  not  qualities  inherent  in  the  bodies  themselves,  by  which  they  immediately  affect      410. 
our  sense,  but  are  mere  consequences  of  that  peculiar  disposition  of  the  particles  of  each  body,  by  which  it  is  Colours  not 
enabled  more  copiously  to  reflect  the  rays  of  one  particular  colour,  and  to  transmit,  or  stifle,  or,  as  it  is  called  in  inherent  in 
Optics,  absorb  the.  others.  bodies. 

Such  is  the  Newtonian  doctrine  of  the  origin  of  'olours.     Every  phenomenon  of  optics  conspires  to  prove  its      41  j 
justice.     Perhaps  the  most  direct  and  satisfactory  proof  of  it  is  to  be  found  in  the  simple  fact,  that  every  body,  Proved  by 
indifferently,  whatever  be  its  colour  in  white  light,  when  exposed  in  the  prismatic  spectrum,  appears  of  the  colour  experiment 
appropriate  to  that  part  of  the  spectrum  in  which  it  is  placed ;  but  that  its  tint  is  incomparably  more  vivid  and 
full  when  laid  in  a  ray  of  a  tint  analogous  to  its  hue  in  white  light,  than  in  any  other.     For  example,  vermillion 
placed  in  the  red  rays  appears  of  the  most  vivid  red  ;  in  the  orange,  orange  ;  in  the  yellow,  yellow,  but  less  bright 
In  the  green  rays,  it  is  green  ;  but  from  the  great  inaptitude  of  vermillion  to  reflect  green  light,  it  appears  dark 
and  dull ;  still  more  so  in  the  blue ;  and  in  the  indigo  and  violet  it  is  almost  completely  black.     On  the  other 
hand,  a  piece  of  dark  blue  paper,  or  Prussian  blue,  in  the  indigo  rays  has  an  extraordinary  richness  and  depth  of 
blue  colour.     In  the  green  its  hue  is  green,  but  much  less  intense ;  while  in  the  red  rays  it  is  almost  entirely 
black.     Such  are  the  phenomena  of  pure  and  intense  colours ;    but  bodies  of  mixed  tints,  as  pink  or  yellow 
paper,  or  any  of  the  lighter  shades  of  blue,  green,  brown,  &c.,  when  placed  in  any  of  the  prismatic  rays,  reflect 
them  in  abundance,  and  appear,  for  the  time,  of  the  colour  of  the  ray  in  which  they  are  placed. 

Refraction  by  a  prism  affords  us  the  means  of  separating  a  ray  of  white  light  into  the  rays  of  different  refran-       412. 
gibility  of  which  it  consists,  or  of  analyzing  it.     But  to  make  the  analysis  complete,  and  to  insulate  a  ray  of  any  Precautions 
particular  refrangibility  in  a  state  of  perfect  purity,  several  precautions  are  required,  the  chief  of  which  are  as  •«  insure  the 
follows:    1st.  The  beam  of  light  to  be  analyzed  must  be  very  small,  as  nearly  as  possible  approaching  to   a  mC0r^'cll°" 
mathematical  ray ;  for  if  A  B,  a  b  be  a  beam  of  parallel  rays  of  any  sensible  breadth  (fig.  89)  incident  on  the  Of  a  ray. 
prism  P,  the  extreme  rays  A  B,  a  b  will  each  be  separated  by  refraction  into  spectra  G  B  H  and  g  b  h :  B  G,  bg  Fig.  89. 
being  the  violet,  and  B  H,  6  A  the  red  rays  of  each  respectively ;  and  since  A  B,  a  6  are  parallel,  therefore  C  G 


408  LIGHT. 

Light,  and  eg-  will  be  so,  and  also  D  H  and  d h.  Hence  the  red  ray  D  H  from  B  will  intersect  the  violet  eg  from  b, 
x—-s/— .^  in  some  point  F  behind  the  prism  ;  and  a  screen  E  Ff  placed  at  F  will  have  the  point  F  illuminated  by  a  red 
t.  Small-  ray  from  B,  and  a  violet  one  from  6  ,•  and  therefore  (as  is  easily  seen)  by  all  the  rays  intermediate  between  the 
\*fd  t  r  an(^  v'°'e'>  from  points  between  B  and  6.  F  therefore  will  be  white.  If  the  screen  be  placed  nearer  the 
pencil.  prism  than  F,  as  at  K  L  k  I,  it  is  clear  that  from  any  point  between  L  and  k  lines  drawn  parallel  to  K  C,  D  L,  to 
any  intermediate  direction,  will  fall  between  C  and  c,  D  and  d,  &c.,  respectively;  and  therefore  that  every  point 
between  L  and  k  will  receive  from  some  point  or  other  of  the  surface  C  d  of  the  prism  a  ray  of  each  colour, 
and  will  therefore  be  white.  Again,  any  point  as  x  between  k  and  I  can  receive  no  violet  ray,  nor  any  ray  of  the 
spectrum  whose  angle  of  deviation  is  greater  than  180°  —  abx;  for  such  ray  to  reach  x  must  come  from  a  part 
of  the  prism  below  b,  which  is  contrary  to  the  supposition  of  a  limited  beam  A  B,  a  5  ;  but  all  rays  whose 
angle  of  deviation  is  less  than  180° —  abx,  will  reach  x  from  some  part  or  other  of  the  surface  B  D.  Hence 
the  colour  of  the  portion  kl  of  the  image  on  the  screen  will  be  white  at  k,  pure  red  at  /,  and  intermediate 
between  white  and  red,  or  a  mixture  of  the  least  refrangible  rays  of  the  spectrum  at  any  intermediate  point ; 
and,  in  the  same  manner,  the  portion  K  L  will  be  white  at  L,  violet  at  K,  and  at  any  intermediate  point  will  have 
a  colour  formed  by  a  mixture  of  a  greater  or  less  portion  of  the  more  refrangible  end  of  the  spectrum.  If  the 
screen  be  removed  beyond  F,  as  into  the  situation  G  g  H  h,  the  white  portion  will  disappear,  no  point  between  g 
and  H  being  capable  of  receiving  any  ray  whose  angle  of  deviation  is  between  180° —  a  b  g  and  180  —  a  b  H. 
We  may  regard  the  whole  image  G  A  as  consisting  of  an  infinite  number  of  spectra  formed  by  every  elementary 
ray  of  which  the  beam  A  B  a  b  is  composed,  overlapping  each  other,  so  that  the  end  of  each  in  succession  projects 
beyond  that  of  the  foregoing.  The  fewer,  therefore,  there  are  of  these  overlapping  spectra,  or  the  smaller  the 
breadth  of  the  incident  beam,  the  less  will  be  the  mixture  of  rays  so  arising,  and  the  purer  the  colours.  Removal 
of  the  screen  to  a  greater  distance  from  the  prism,  evidently  produces  the  same  effect  as  diminution  of  the  size 
of  the  beam ;  for  while  each  colour  occupies  constantly  the  same  space  on  the  screen  (for  G  g  =  K  k)  the  whole 
spectrum  is  diffused  over  a  larger  space  as  the  screen  is  removed,  by  the  divergence  of  its  component  rays  of 
different  colours,  and  therefore  the  individual  colours  must  of  necessity  be  continually  more  and  more  separated 
from  each  other. 

413.  Sndly.  Another  source  of  confusion  and  want  of  perfect  homogeneity  in  the  colours  of  the  spectrum   is  the 
2nd.  Small  angular  diameter  of  the  sun  or  other  luminary,  even  when  the  aperture  through  which  the  beam  is  admitted  is 
vereence  of  eyer  so  much  diminished-     For  let  s  T  (fig-  90)  be  the  sun>  whose  rays  are  admitted  to  the  prism  ABC  through 
the  pencil.    a  verY  sma'l  hole  O  in  a  screen  placed  close  to  it.     The  beam  will  be  dilated  by  refraction  into  the  spectrum  v  r. 
Fig.  90.       Now,  if  we  consider  only  the  rays  of  one  particular  kind,  as  the  red,  and  regard  all  the  rest  as  suppressed,  it  is 

clear  that  a  red  image  r  of  the  sun  will  be  formed  by  them  alone  on  the  screen  ;  the  rays  from  every  point  of 
the  sun's  disc  crossing  at  O,  and  pursuing  (after  refraction)  different  courses.  If  the  prism  be  placed  in  its 
situation  of  minimum  deviation,  which  at  present  we  will  suppose,  this  image  will  be  a  circle,  and  it  and  the  sun 
will  subtend  equal  angles  at  O.  In  like  manner,  the  violet  rays  (considered  apart  from  the  red)  will  form  a 
circular  violet  image  of  the  sun,  at  v,  by  reason  of  their  greater  refrangibility ;  and  every  species  of  ray,  of 
intermediate  refrangibility,  will  form,  in  like  manner,  a  circular  image  between  r  and  v.  The  constitution  of  the 
spectrum  so  arising  will  therefore  be  as  in  fig.  91,  a,  being  an  assemblage  of  images  of  every  possible  refrangi- 
bility superposed  on  and  overlapping  each  other.  Now,  if  we  diminish  the  angular  diameter  of  the  sun  or 
luminary,  each  of  these  images  will  be  proportionally  diminished  in  size  ;  but  their  number,  and  the  whole 
extent  over  which  they  are  spread,  will  remain  the  same.  They  will  therefore  overlap  less  and  less,  (as  in 
Fig.  91.  fig.  91,  6,  c,-)  and  if  the  luminary  be  conceived  reduced  to  a  mere  point  (as  a  star)  the  spectrum  will  consist  of 
a  line  d  composed  of  an  infinite  number  of  mathematical  points,  each  of  a  perfectly  pure  homogeneous  light. 

414.  There  are  several   ways  by  which  the  angular  diameter,  or  the  degree  of  divergence  of  the  incident  beam  may 
Experimen-  be  diminished.     Thus,  first,  we  may  admit  a  sunbeam  through  a  small  hole,  as  A,  in  a  screen,  and  receive  the 
tal  methods  divergent  cone  of  rays  behind  it  on  another  screen  B,  (fig.  7,)  at  a  considerable  distance,  having  another  small 
homoee'"'""  nole  B  to  let  Pass>  not  tne  whole,  but  only  a  small  portion  of  the  sun's  image.     The  beam  B  C,  so  transmitted, 
neous  pris-  W'U  manifestly  have  a  degree  of  divergence  less  than  that  of  the  beam  immediately  transmitted  from  A  in  the 
matic  rays,    proportion  of  the  diameter  of  the  aperture  B  to  the  diameter  of  the  sun's  image  on  the  screen  B. 

F'g-  7-  Another  and  much  more  commodious  method  is  to  substitute  for  the    sun  its  image  formed  in  the  focus 

415.  of  a  convex  lens  of  short  focus.     This  image  is  of  very  small  dimensions,  its  diameter  being  equal  to  focal 
Fig.  92.       length  of  the  lens  x  sine  of  sun's  angular  diameter,  (or  sine  of  30',  which  is  about  one  114th  part  of  radius,) 

so  that  a  lens  of  an  inch  focus  concentrates  all  the  rays  which  fall  on  it  within  a  circle  of  about  the  114th 
of  an  inch  in  diameter,  which,  for  this  purpose,  may  be  regarded  as  a  physical  point.  The  disposition  of  the 
apparatus  is  as  represented  in  fig.  92.  The  rays  converged  by  the  lens  L  to  F,  afterwards  diverge  as  if  they 
emanated  from  an  intensely  bright  luminous  point  placed  at  F,  and  a  screen  with  a  small  aperture  O  being 
placed  at  a  distance  from  it,  and  .close  behind  it  the  prism  ABC,  the  spectrum  r  v  may  be  received  on  a  screen 
again  placed  at  a  considerable  distance  behind  the  prism,  each  of  whose  points  will  be  illuminated  by  rays  of  a 
very  high  degree  of  purity  and  homogeneity,  and  by  diminishing  the  focal  length  of  the  lens,  and  the  aperture 
O,  and  increasing  the  distance  F  O,  or  O  r,  this  may  be  carried  to  any  extent  we  please.  It  should,  however, 
be  remarked,  that  the  intensity  of  the  purified  ray,  and  the  quantity  of  homogeneous  light  so  obtained,  are 
diminished  in  the  same  ratio  as  the  purity  of  the  ray  is  increased. 

416.  A  third  method  of  obtaining  a  homogeneous  beam  is  to  repeat  the  process  of  analysis  on  a  ray  as  nearly 
Fig.  93.       pure    as    can  be    conveniently  obtained  by  refraction  through    a    single    prism.     Thus,  in  fig.  93,  V  R,    the 

spectrum  formed  by  a  first  refraction  at  the  prism  A,  is  received  on  a  screen  which  intercepts  the  whole  of 
it,  except  that  particular  colour  we  wish  to  insulate  and  purify,  which  is  allowed  to  pass  through  an  aperture 
M  N ;  behind  this  is  placed  another  prism  B,  so  as  to  refract  this  beam  a  second  time.  If  then  the  portion 


LIGHT.  409 

I.i.jlit.      M  N  were  already  perfectly  pure,  it  would  pass  the  second  prism  without  undergoing1  any  further  separation  ;     Part  II. 
»-v— «^  but  if  there  be  (as  there  always  will)  other  rays  mixed  with  it,  these  will  be  dilated  by  the  subsequent  refraction  v-~v^"' 
into  a  second  spectrum  vr  of  faint  light,  with  a  much  brighter  portion  mn  in  the  midst;  and  if  the  rest  of  the 
rays  be  intercepted,  and  this  portion  only  allowed  to  pass  through  an  aperture,  the  emergent  beam  mp  will  be 
much  more  homogeneous  than  before  its  incidence  on  the  second  prism, — and  in  proportion  as  the  distance  be- 
tween the  second  prism  and  the  screen  is  increased,  the  purity  of  the  ray  obtained  will  be  greater. 

Another  source  of  impurity  in  the  prismatic  rays  is  the  imperfection  of  the  materials  of  our  ordinary  prisms,      417 
which  are  full  of  striae  and  veins,  which  disperse  the  light  irregularly,  and  thus  confound  together  in  the  spectrum  Imperfec- 
rays  which  properly  belong  to  different  parts  of  it.     Those  who  are  not  fortunate  enough  to  possess  glass  prisms  tion  of 
free  from  this  defect  (which  are  very  rare,  and  indeed  hardly  to  be  procured  for  any  price)  may  obviate  the  in-  Prisms) 
convenience  by  employing  hollow  prisms  full  of  water,  or,  rather,  any  of  the  more  dispersive  oils.     A  great  part  •   w     ] 
of  the  inconvenience  arising  from  a  bad  prism  may,  however,  be  avoided  by  transmitting  the  rays  as  near  the 
edge  of  it  as  possible,  so   as  to  diminish  the  quantity  of  the  material  they  have  to  pass  through,  and  therefore 
their  chance  of  encountering  veins  and  striae  in  their  passage. 

When  every  care  is  taken  to  obtain  a  pure  spectrum ;  when  the  divergence  of  the  incident  beam  is  extremely      415. 
small,  and  its  dimensions  also  greatly  reduced  ;  when  the  prism  is  perfect,  and  the  spectrum  sufficiently  elon-  Fixed  linei 
gated  to  allow  of  a  minute  examination  of  its  several  parts,  some  very  extraordinary  facts  have  been  observed  '"  'he 
respecting  its  constitution.     They  were  first  noticed  by  Dr.  Wollaston,  in  a  Paper  published  by  him  in  the  Phil.  sPectrum- 
Trans.,  1802  ;    and   have  since   been  examined  in  full  detail,  and  with  every  delicacy  and  refinement  which 
the  highest  talents  and  the  most  unlimited  command  of  instrumental  aids  could  afford,  by  the  admirable  and 
ever-to-be-lamented  Fraunhofer.     It  does  not  appear  that   the  latter  had  any  knowledge   of  Dr.  Wollaston's 
previous  discovery,  so  that  he  has,  in  this   respect,  the  full  merit  of  an  independent  inventor.     The  facts  are 
these  :    The  solar  spectrum,  in  its  utmost  possible  state  of  purity  and  tenuity,  when  received  on  a  white  screen, 
or  when  viewed  by  admitting  it  at  once  into  the  eye,  is  not  an  uninterrupted  line  of  light,  red  at  one  end  and 
violet  at  the   other,  and  shading  away  by  insensible  gradations  through  every  intermediate  tint  from  one  to  the 
other,  as  Newton  conceived  it  to  be,  and  as  a  cursory  view  shows  it.     It  is  interrupted  by  intervals  absolutely 
dark ;   and  in  those  parts  where  it  is  luminous,  the  intensity  of  the  light  is  extremely  irregular  and  capricious, 
and  apparently  subject  to  no  law,  or  to  one  of  the  utmost  complexity.     In  consequence,  if  we  view  a  spectrum 
formed  by  a  narrow  line  of  light  parallel  to  the  refracting  edge  of  the  prism,  (which  affords  a   considerable 
breadth  of  spectrum  without  impairing  the  purity  of  the  colours,  being,  in  fact,  an  assemblage  of  infinitely  narrow 
linear  spectra  arranged  side  by  side,)  instead  of  a  luminous  fascia  of  equable  light  and  graduating  colours,  it 
presents  the  appearance  of  a  striped   riband,  being   crossed  in  the  direction  of  its  breadth  by  an  infinite  multi- 
tude of  dark,  and  by  some  totally  black  bands,  distributed  irregularly  throughout  its  whole  extent.     This  irregu- 
larity, however,  is  not  a  consequence  of  any  casual  circumstances.     The  bands  are  constantly  in  the  same  parts 
of  the  spectrum,  and  preserve  the  same  order  and  relations  to  each  other ;   the  same  proportional  breadth  and 
degree  of  obscurity,  whenever  and  however  they  are  examined,  provided  solar  light  be  used,  and  provided  the 
prisms  employed  be  composed  of  the  same  material :  for  a  difference  in  the  latter  particular,  though  it  causes  no 
change  in  the  number,  order,  or  intensity  of  the  bands,  or  their  places  in  the  spectrum,  as  referred  to  the  several 
colours  of  which  it  consists,  yet  causes  a  variation  in  their  proportional  distances  inter  se,  of  which  more  here- 
after.    By  solar  light  must  be  understood,  not  merely  the  direct  rays  of  the  sun,  but  any  rays  which  have  the 
sun  for  their  ultimate  origin  ;  the  light  of  the  clouds,  or  sky,  for  instance ;  of  the  rainbow  ;  of  the  moon,  or  of 
the  planets.     All  these  lights,  when  analyzed  by  the  prism,  are  found  deficient  in  the  identical  rays  which   are 
wanting  in  the  solar  spectrum  ;  and  the  deficiency  is  marked  by  the  same  phenomenon,  viz.  by  the  occurrence  of 
the  same  dark  bands  in  the  same  situations  in  spectra  formed  by  these  several  lights.     In  the  light  of  the  stars, 
on  the  other  hand,  in  electric  light,  and  that  of  flames,  though  similar  bands  are   observed  in  their  spectra, 
yet  they  are  differently  disposed;  and  the  spectrum  of  each  several  star,  and  each  flame,  has  a  system  of  bands 
peculiar  to  itself,   and  characteristic  of   its  light,  which  it  preserves    unalterably  at    all  times,  and  under  all 
circumstances. 

Fig.  94  is  a  representation  of  the  solar  spectrum  as  laid  down  minutely  by  Fraunhofer,  from  micrometrical      419. 
measurement,  and  as  formed  by  a  prism  of  his  own  incomparable  flint  glass.     Only  the  great  number  of  small  Fig.  94. 
bands  observed  by  him  (upwards  of  500  in  number)  have  been  omitted,  to  avoid  confusing  the  figure.     Of  these 
bands,  or,  as  he  terms  them,  "  fixed  lines"  in  the  spectrum,  he  has  selected  seven,  (those  marked  B,  C,  D,  E,  F, 
G,  H,)  as  terms  of  comparison,  or  as  standard  points  of  reference  in  the  spectrum,  on  account  of  their  distinct- 
ness, and  the  facility  with  which  they  may  be  recognised.     Of  these,  B  lies  in  the  red  portion  of  the  spectrum, 
near  the  end;  C  is  farther  advanced  in  the  red;  D  lies  in  the  orange,  and  is  a  strong  double  line  easily  recog- 
nised ;  E  is  in  the  green  ;  F  in  the  blue  ;    G   in   the  indigo  ;   and  H  in  the  violet.     Besides  these,  there  are 
others  very  remarkable ;  thus  6  is  a  triple  line  in  the  green,  between  E  and  F,  consisting  of  three  strong  lines, 
of  which  two  are  nearer  each  other  than  the  third,  &c. 

The  definiteness  of  these  lines,  and  their  fixed  position,  with  respect  to  the  colours  of  the  spectrum,— in       420. 
other  words,  the  precision  of  the  limits  of  those  degrees  of  refrangibility  which  belong  to  the  deficient  rays  Utility  of 
of  solar  light, — renders  them  invaluable  in  optical  inquiries,  and  enables  us  to  give  a  precision  hitherto  unheard  l!'e  fixei1 
of  to  optical  measurements,  and  to  place  the  determination  of  the  refractive  powers  of  media  on  the  several  rays  ||"j  1"  °f 
almost  on  the  same  footing,  with  respect  to  exactness,  with  astronomical   observations.      Fraunhofer,  in  his  ^nations'' 
various  essays,  has  made  excellent  use  of  them  in  this  respect,  as  we  shall  soon  have  occasion  to  see. 

To  see  these  phenomena,  we  must  place  the  refracting  angle  of  a  very  perfect  prism  parallel  to  a  very  small       421 
linear  opening  through  which  a  sunbeam  is  admitted  ;  or,  in  place  of  an  opening,  we  may  employ  a  glass 
cylinder,  or  semi-cylinder  of  small  radius,  to  bring  the  rays  to  a  linear  focus  behind  and  parallel  to  it,  from 
VOL.  iv.  3  H 


410 


LIGHT. 


Light. 


First  me- 
thod of  ex- 
hibiting the 
fixed  lines. 


422. 

Second 

method. 


Fi«.  95. 


423. 

Third 

method. 


Fig    96. 
424. 

Colours  of 
the  spec 
'.rum. 


which  the  rays  diverge,  as  from  a  fine  luminous  line,  in  the  manner  described  in  Art.  415  for  a  lens.  If  now  the 
eye  be  applied  close  behind  the  prism,  the  line  will  be  seen  dilated  into  a  broad  coloured  band,  consisting  of  the  •> 
prismatic  colours  in  their  order ;  and  if  the  prism  be  good,  and  carefully  placed  in  its  situation  of  minimum 
deviation,  and  of  sufficiently  large  refracting  angle  to  give  a  broad  spectrum,  some  of  the  more  remarkable  of 
the  fixed  lines  will  be  seen  arranged  parallel  to  the  edges  of  the  spectrum,  especially  the  lines  D  and  F,  the 
former  of  which  appears,  in  this  way  of  viewing  it,  to  form  a  separation  between  the  red  and  the  yellow.  If 
the  light  of  the  sun  be  too  bright,  so  as  to  dazzle  the  eye,  any  narrow  line  of  common  daylight  (as  the  slit 
between  two  nearly  closed  window-shutters)  may  be  substituted.  This  was  the  mode  in  which  the  fixed  lines 
were  first  discovered  by  Dr.  Wollaston. 

But  it  is  difficult  and  requires  acute  sight  to  perceive,  in  this  manner,  any  but  the  most  conspicuous  lines. 
The  reason  is,  their  very  small  angular  breadth ;  which,  in  the  largest  of  them,  can  scarcely,  under  any  circum- 
stances, exceed  half  a  minute,  and  in  the  smaller  not  more  than  a  few  seconds.  They  require,  therefore,  to  be 
magnified.  This  may  be  done  by  a  telescope  interposed  between  the  eye  and  the  prism,  in  the  manner  repre- 
sented in  fig.  95,  in  which  L  /  is  the  line  of  light,  from  which  rays,  diverging  in  all  directions,  fall  on  the  prism 
ABC,  are  refracted  by  it,  and  after  refraction  are  received  on  the  object-glass  D  of  the  telescope.  This  object- 
glass,  it  should  be  observed,  must  be  of  that  kind  denominated  achromatic,  to  be  presently  described,  (see  Index,) 
and  of  which  it  need  only  be  here  said,  that  it  is  so  constructed  as  to  be  capable  of  bringing  rays  of  all  colours  to  foci 
at  one  and  the  same  distance  from  the  glass.  Now,  if  we  consider  only  rays  of  any  one  degree  of  refrangibility 
(the  extreme  red,  for  instance)  the  pencils  diverging  from  every  point  of  L I  will,  after  refraction  at  the  two 
surfaces  of  the  prism,  diverge  from  corresponding  points  of  an  image  L'/'  situated  in  the  direction  from  the 
base  towards  the  vertex  of  the  prism.  Rays  of  any  greater  refrangibility  will,  after  refraction  at  the  prism,  diverge 
from  a  linear  image  L," I"  parallel  to  L'/',  but  farther  from  the  original  line  L/.  Thus  the  white  line  L  I  will, 
after  refraction  at  the  prism,  have  for  its  image  the  coloured  rectangle  L'  li"l'  I",  which  will  be  viewed  through 
the  telescope  as  if  it  were  a  real  object.  Now  every  vertical  line  of  this  parallelogram  will  form  in  the  focus  of 
the  object-glass  a  corresponding  vertical  image  of  its  own  colour  ;  and  the  object-glass  being  achromatic,  all 
these  images  are  equidistant  from  it,  so  that  the  whole  image  of  the  parallelogram  I/  I"  will  be  a  similar  coloured 
parallelogram,  having  its  plane  perpendicular  to  the  axis  of  the  telescope.  This  will  be  viewed  as  a  real  object 
through  the  eye-glass,  and  the  spectrum  will  thus  be  magnified  as  any  other  object  would  be,  according  to  the 
power  of  the  telescope,  (Art.  382.)  With  this  disposition  of  the  apparatus  (which  is  that  employed  by  Fraun- 
hofer)  the  fixed  lines  are  beautifully  exhibited,  and  (if  the  prism  be  perfect)  may  be  magnified  to  any  extent. 
The  slightest  defect  of  homogeneity  in  the  prism,  however,  as  may  be  readily  imagined,  is  fatal.  With  glass 
prisms  of  our  manufacture  it  would  be  quite  useless  to  attempt  the  experiment ;  and  those  who  would  repeat  it 
in  this  country  should  employ  prisms  of  highly  refractive  liquids,  enclosed  in  hollow  prisms  of  good  plate  glass 
The  eye-pieces  of  telescopes,  not  being  usually  achromatic,  a  slight  change  of  focus  is  still  required,  when  the 
lines  in  the  red  and  violet  portions  of  the  spectrum  are  to  be  viewed.  This  (if  an  inconvenience)  might  be 
obviated  by  the  use  of  an  achromatic  eye-piece. 

That  an  actual  image  of  the  spectrum,  with  its  fixed  lines,  is  really  formed  in  the  focus  of  the  object-glass, 
as  described,  may  be  easily  shown,  by  dismounting  the  telescope,  and  receiving  the  rays  refracted  by  the  object- 
glass  on  a  screen  in  its  focus.  This,  indeed,  affords  a  peculiarly  elegant  and  satisfactory  'mode  of  exhibiting  the 
phenomena  to  several  persons  at  once.  An  achromatic  object-glass  of  considerable  focal  length  (6  feet,  for 
instance)  should  be  placed  at  about  twice  its  focal  length  from  the  line  of  light,  and  (the  prism  being  placed 
immediately  before  the  glass)  the  image  will  be  formed  at  about  the  same  distance,  12  feet  behind  it,  (f=  L  4- 
D;  L  =  £ ;  D  =  — TV;  f=  £  —  -A-  =  4-  iV)  and  being  received  on  a  screen  of  white  paper  or  emeried  glass 
may  be  examined  at  leisure,  and  the  distances  of  the  lines  from  each  other,  &c.  measured  on  a  scale.  But  by 
far  the  best  methods  of  performing  these  measurements  are  those  practised  by  Fraunhofer,  viz.  the  adaptation 
of  a  micrometer  to  the  eye-end  of  the  telescope,  (see  Micrometer,  in  a  subsequent  part  of  this  Article,)  for  ascer- 
taining the  distances  of  the  closer  lines;  and  the  giving  the  axis  of  the  telescope,  together  with  the  prism  which 
is  connected  with  it,  a  motion  of  rotation  in  a  horizontal  plane,  the  extent  of  which  is  read  off  by  verniers  and 
microscopes  on  an  accurately  graduated  circle,  in  the  same  way  as  in  astronomical  observations.  The  apparatus 
employed  by  him  for  this  purpose,  and  which  is  applicable  to  a  variety  of  useful  purposes  in  optical  researches, 
is  represented  in  fig.  96. 

The  fixed  lines  in  the  spectrum  do  not  mark  any  precise  limits  between  the  different  colours  of  which  it 
consists.  According  to  Dr.  Wollaston,  (Phil.  Trans.,  1802,)  the  spectrum  consists  of  only  four  colours,  red, 
green,  blue,  and  violet ;  and  he  considers  the  .narrow  line  of  yellow  visible  in  it  in  his  mode  of  examination 
already  described  (looking  through  a  prism  at  a  narrow  line  of  light  with  the  naked  eye)  as  arising  from  a 
mixture  of  red  and  green.  These  colours,  too,  he  conceives  to  be  well  defined  in  the  spaces  they  occupy,  not 
graduating  insensibly  into  each  other,  and  of,  sensibly,  the  same  tint  throughout  their  whole  extent.  We  confess 
we  have  never  been  able  quite  satisfactorily  to  verify  this  last  observation,  and  in  the  experiments  of  Fraunhofer, 
(which  we  had  the  good  fortune  to  witness,  as  exhibited  by  himself  at  Munich,)  where,  from  the  perfect  distinctness 
of  the  finest  lines  in  the  spectrum,  all  idea  of  confusion  of  vision,  or  intermixture  of  rays  is  precluded,  the  tints 
are  seen  to  pass  into  each  other  by  a  perfectly  insensible  gradation  ;  and  the  same  thing  may  be  noticed  in  the 
coloured  representations  of  the  spectrum  published  in  the  first  essay  of  that  eminent  artist,  and  executed  by 
himself  with  extraordinary  pains  and  fidelity.  The  existence  of  a  pale  straw  yellow,  not  of  mere  linear  breadth, 
but  occupying  a  very  sensible  space  in  the  spectrum,  is  there  very  conspicuous,  and  may  also  be  satisfactorily 
shown  by  other  experiments  to  be  hereafter  described,  when  we  come  to  speak  of  the  absorption  of  light.  In 
short,  (with  the  exception  of  the  fixed  lines,  which  Newton's  instrumental  means  did  not  enable  him  to  see,)  the 
spectrum  is,  what  that  illustrious  philosopher  originally  described  it,  a  graduated  succession  of  tints,  in  which  all 


Part  II. 


LIGHT.  411 

the  seven  colours  he  enumerates  can  be  distinctly  recognised,  but  shading  so  far  insensibly  into  each  other  that  a  Part  II. 
positive  limit  between  them  can  be  nowhere  fixed  upon.  Whether  these  colours  be  really  compound  or  not,  whether  v— — v—* 
some  other  mode  of  analysis  may  not  effect  a  separation  depending  on  some  other  fundamental  difference  between 
the  rays  than  that  of  the  degree  of  their  refrangibility,  is  quite  another  question,  and  will  be  considered  more  at 
large  hereafter.  At  present  it  may  be  enough  to  remark,  that  all  probability,  drawn  from  everyday  experience, 
is  in  favour  of  this  idea,  and  leads  us  to  believe  that  orange,  green,  and  violet  are  mixed  colours  ;  and  red, 
yellow,  and  blue,  original  ones ;  the  former  we  everyday  see  imitated  by  mixtures  of  the  latter,  but  never  vice 
versa.  This  doctrine  has  been  accordingly  maintained  by  Mayer,  in  a  curious  Tract  published  among  his  works. 
(See  the  Catalogue  of  Optical  Writers  at  the  end  of  this  Article.)  A  very  different  doctrine  has,  however,  been 
advanced  by  Dr.  Young,  (Lectures  on  Natural  Philosophy,  i.  441,)  in  which  he  assumes  red,  green,  and  violet, 
as  the  fundamental  colours.  The  respective  merits  of  these  systems  will  be  considered  more  at  large  hereafter. 
(See  Index,  Composition  of  Colours.) 


Media,  as  we  have  seen,  differ  very  greatly  in  their  refractive  power,  or  in  the  degree  in  which  prisms  of  one  and  the      425. 
same  refracting  angle  composed  of  different  substances,  deflect  the  rays  of  light.    This  was  known  to  the  optical  phi-  M?dia; 
losophers  who  preceded  Newton.    This  great  man,  on  establishing  the  general  fact,  that  one  and  the  same  medium  dj,  g^"ve 


refracts  differently  the  differently  coloured  rays,  might  naturally  have  been  led  to  inquire  experimentally  whether  __ 
the  amount  of  this  difference  of  action  were  the  same  for  all  media.  He  appears  to  have  been  misled  by  an  acci-  ' 
dental  circumstance  in  the  conduct  of  an  experiment,  in  which  the  varieties  of  media  in  this  respect  ought  to  have 
struck  him,*  and  in  consequence  adopted  the  mistaken  idea  of  a  proportional  action  of  all  media  on  the  several  homo- 
geneous rays.  Mr.  Hall,  a  gentleman  of  Worcestershire,  was  the  first  to  discover  Newton's  mistake ;  and  having 
ascertained  the  fact,  of  the  different  dispersive  powers  of  different  kinds  of  glass,  applied  his  discovery  successfully 
to  the  construction  of  an  achromatic  telescope.  His  invention,  however,  was  unaccountably  suffered  to  fall  into 
oblivion,  (though  it  is  said  that  he  made  several  such  telescopes,  some  of  which  still  exist,)  and  the  fact  was 
re-discovered  and  re-applied  to  the  same  great  purpose  by  Mr.  Dollond,  a  celebrated  optician  in  London,  on 
the  occasion  of  a  discussion  raised  on  the  subject  by  some  a  priori  and  paradoxical  opinions  broached  by 
Euler. 

If  a  prism  of  flint  glass  and  one   of  crown,  of  equal  refracting  angles,  be  presented  to  two  rays  of  white      426. 
light,  as  A  B  C,  a  be,  (fig.  97  ;)  S  C  and  sc  being  the  incident  rays,  C  R,  C  V  the  red  and  violet  rays  refracted  Differences 
by  the  flint,  and  or,  cv  those  refracted  by  the  crown  ;  it  is  observed,  first,  that  the  deviation  produced  in  either  of  disper- 
the  red  or  violet  ray  by  the  flint  glass,  is  much  greater  than  that  produced  by  the  crown  ;  secondly,  that  the  angle  s']Ca"ngj" 
RC  V,  over  which  the  coloured  rays  are  dispersed  by  the  flint  prism,  is  also  much  greater  than  the  angle  rev,  ^"97 
over  which  they  are  dispersed  by  the  crown  ;  and,  thirdly,  that  the  angles  R  C  V,  r  CD,  or  the  angles  of  disper- 
sion, are  not  to  each   other  as  Newton  supposed  them  to  be,  in  the  same  ratio  with  the  angles  of  deviation 
T  C  R,  tcr,  but  in  a  much  higher  ratio ;  'the  dispersion  of  the  flint  prism  being  much  more  than  in  proportion 
to  the  deviation  produced  by  it.    And  if,  instead  of  taking  the  angles  of  the  prism  equal,  the  refracting  angle  of  the 
crown  prism  be  so  increased  as  to  make  the  deviation  of  the  red  ray  equal   to  that  produced  by  the  flint  prism, 
the  deviation   of  the  violet  will  fall   considerably  short  of  such  equality.     In  consequence  of  this,   if  the  two 
prisms  be  placed  close  together,  with  their  edges  turned  opposite  ways,  as  in  fig.  98,  so  as  to  oppose  each  other's  ^'S-  98. 
action,  the  red  ray,  being  equally  refracted  in  opposite  directions,  will  suffer  no  deviation  ;  but  the  violet  ray, 
being  more  refracted  by  the  flint  than  by  the  crown  prism,  will,  on  the  whole,  be  bent  towards  the  thicker  part 
of  the  flint  prism,  and  thus  an  uncorrected  colour  will  subsist,  though  the  refraction  (for  one  ray,  at  least)  is 
corrected.     Vice  versa,  if  the  dispersion  be  corrected,  that  is,  if  the  refracting  angle  of  the  crown  prism,  acting 
in  opposition  to  the  flint,  be  so  further  increased  as  to  make  the  difference  of  the  deviations  of  the  red  and  violet 
rays  produced  by  it  equal  to  the  difference  of  their  deviations  produced  by  the  flint,  the  deviation  produced  by 
it  will  now  be  greater  than  that  produced  by  the  flint ;    and  the  total  deviation,  produced  by  both  prisms  acting 
together,  will  now  be  in  favour  of  the  crown. 

By  such  a  combination  of  two  prisms  of  different  media  a  ray  of  white  light  may  therefore  be  turned  aside      427. 
considerably  from  its  course,  without  being  separated  into  its  elementary  coloured  rays.     It  is  manifest,  that  (sup-  Rf^'ion 
posing  the  angles  of  the  prisms  small,  and  that  both  are  placed  in  their  positions  of  minimum   deviation)  the  ™j|2"^  ** 
deviations  to  produce  this  effect  must  be  in  the  inverse  ratio  of  the  dispersive  powers  of  the  two  media ;    for  jnt0  colours, 
supposing  ft,  ft'  to  be  the  refractive  indices  of  the  prisms  for  extreme  red  rays,  and  /i  +  S  ft,  ft'  +  S  fif  for  extreme 
violet,  A  and  A'  their  refracting  angles,  and  D  and  D'  their  deviations,  we  have,  generally,  in  the  position  of 
minimum  deviation 


,  whence  c  fa  .  sin  —  =  £  c  D  .  cos 


A'  A'  +  D' 

.  sin  —  —  =  sin 


whence,  since  the  prisms  oppose  each  other, 


*  He  counteracted  the  refraction  of  a  glass,  by  a  water  prism.  There  ought  to  have  been  a  residuum  of  uncorrected  colour;  but, 
unluckily,  he  had  mixed  sugar  of  lead  with  the  water  to  increase  its  refraction,  and  the  high  dispersive  power  of  the  salts  of  lead  (of  which 
of  course,  he  could  not  have  the  least  suspicion)  thus  robbed  him  of  one  of  the  greatest  discoveries  in  physical  optics. 

3  H2 


412 


LIGHT. 


Light. 


I  (D  -  D')  = 


_A_ 

2 


A' 


Part  IL 


cos 


Putting  this  equal  to  zero,  we  have 


sin^  A  cos  £  (A  +  D) 

'  sin  J  A'  =      cos  i  (A'  +  IX)  5 


and,  eliminating  sin  £  A  and  sin  f  A'  from  this,  by  means  of  the  two  original  equations  from  which  we  set  out, 
we  get 

a/.  p.'          cos  ^  (A  +  D)  sin^i  (AM^DO_  _     tan  $  (A7  +  DQ 

a  /    '     ~JT  ''  '    cos  £  (A'  +  D')    :  :    sin  i  (A  +  D)  tan  £  (A  +  D) 

Now  if  we  call  p,  p'  the  dispersive  powers  of  the  media,  or  the  proportional  parts  of  the  whole  refractions  of  the 
extreme  red  ray,  to  which  the  dispersion  is  equal,  we  shall  have 


p  = 


and  -±-r  = 


so  that 
P 


a/ 

A' 


f!  —  1  tan  \  (A'  +  DQ  ft'  —  I       _s 

/*  —  1  tan  £  (A  +  D)  /»  —  1         sin  £  A 

Such  is  the  strict  formula,  which,  when  A  and  A'  are  verv  small,  becomes 


A)4 


P        -    &  ~  ^  A' 


(p.  -  1)  A 


D  D' 

or,  since  (/*  -  1)  A  =  D,  and  (/  -  1)  A1  =  D';        ~V  =  — . 


428. 
Dispersive 
powers  com 
pared  by 
experiment. 


429. 

Coloured 
fringes  bor- 
dering ob- 
jects seen 
through 
prisms  ex- 
plained. 
Fig.  99. 


The  formula  just  obtained,  furnishes  us  with  an  experimental  method  of  determining  the  ratio  of  the  dispersive 
powers  of  two  media.  For  if  we  can  by  any  means  succeed  in  forming  them  into  two  prisms  of  such  refracting 
angles,  that,  when  placed  in  their  respective  positions  of  minimum  deviation,  a  well  defined  bright  object,  viewed 
through  both,  shall  appear  well  defined  and  free  from  colour  at  its  edges ;  then,  by  measuring  their  angles,  and 
knowing  also  from  other  experiments  their  refractive  indices,  the  equation  (a)  gives  us  immediately  the  ratio  in 
question. 

When  we  view  through  a  prism  any  well  defined  object,  either  much  darker  or  much  lighter  than  the  ground 
against  which  it  is  seen  projected,  as,  for  instance,  a  window  bar  seen  against  the  sky,  its  edges  appear  fringed 
with  colours  and  ill  defined.  The  reason  of  this  may  be  explained  as  follows : 

Let  A  B,  fig.  99,  be  the  section  of  a  horizontal  bar  seen  through  the  prism  P  held  with  its  refracting  edge 
downwards,  and  first  let  us  consider  what  will  be  the  appearance  of  the  upper  edge  B  of  the  object.  Since  we 
see  by  light,  and  not  by  darkness,  the  thing  really  seen  is  not  the  dark  object,  but  the  bright  ground  on  which  it 
stands,  or  the  bright  spaces  B  C,  A  D  above  and  below.  Now  the  bright  space  B  C  above  the  object  being 
illuminated  with  white  light,  will,  after  refraction  at  the  prism,  form  a  succession  of  coloured  images  6  c,  V  c', 
b"  c'',  &c.,  superposed  on  and  overlapping  each  other.  They  are  represented  in  the  figure  as  at  different  dis- 
tances from  P,  but  this  is  only  to  keep  them  distinct.  In  reality,  they  must  be  supposed  to  lie  upon  and  interfere 
with  each  other.  The  least  refracted  6  c  of  these  is  red,  and  the  most  refracted  b"  e"  violet,  and  any  intermediate 
one  (as  b'  d)  of  some  intermediate  colour,  as  yellow  for  instance.  Beyond  b"  no  image  exists,  so  that  the  whole 
space  below  6"  will  appear  dark  to  an  eye  situated  behind  the  prism.  On  the  other  hand,  above  b  the  images  of 
every  colour  in  the  spectrum  coexist,  the  bright  space  6  c  being  supposed  to  extend  indefinitely  above  B.  There- 
fore the  space  above  6  in  the  refracted  image  will  appear  perfectly  white.  Between  6  and  b"  there  will  be  seen, 
first,  a  general  diminution  of  light,  as  we  proceed  from  6  towards  b",  because  the  number  of  superposed  luminous 
images  continually  decreases  ;  secondly,  an  excess  in  all  this  part,  of  the  more  refrangible  rays  in  the  spectrum 
above  what  is  necessary  to  form  white  light,  for  beyond  6  no  red  image  exists,  beyond  6'  no  yellow,  and  so  on  ; 
the  last  which  projects  beyond  all,  at  b",  being  a  pure  unmixed  violet.  Thus  the  light  will  not  only  decrease  in 
intensity,  but  by  the  successive  subtraction  of  more  and  more  of  the  less  refrangible  end  of  the  spectrum  will 
acquire  a  bluer  and  bluer  tint,  deepening  to  a  pure  violet,  so  that  the  upper  edge  of  the  dark  object  will  appear 
fringed  with  a  blue  border,  becoming  paler  and  paler  till  it  dies  away  into  whiteness.  The  reverse  will  happen 
at  the  lower  edge  A.  The  bright  space  A  D  forms,  in  like  manner,  a  succession  of  coloured  images,  a  d,  a'  d', 
a'  d'',  of  which  the  least  deviated  a  d  is  red,  the  most  a"  d"  violet,  and  the  intermediate  ones  of  the  intermediate 
colours.  Therefore  the  point  a,  which  contains  only  the  extreme  red,  will  appear  of  a  sombre  red  ;  a',  which 
contains  all  the  rays  from  red  to  yellow  (suppose),  of  a  lively  orange  red ;  and  in  proportion  as  the  other  images 
belonging  to  the  more  refrangible  end  of  the  spectrum  come  in,  this  tendency  to  an  excess  of  red  will  be  neutralized, 
and  the  portion  beyond  a1',  containing  all  the  colours  in  their  natural  proportions,  will  be  purely  white.  Hence, 
the  lower  edge  of  the  dark  object  will  appear  bordered  with  a  red  fringe,  whose  tint  fades  away  into  whiteness, 
in  the  same  way  as  the  blue  fringe  which  borders  the  upper  edge.  These  fringes,  of  course,  destroy  the  dis- 
tinctness of  the  outlines  of  objects,  and  render  vision  through  a  prism  confused.  The  confusion  ceases,  and 
objects  resume  their  natural  well  defined  outlines,  if  illuminated  with  homogeneous  light,  or  if  viewed  through 
coloured  glasses  which  transmit  only  homogeneous  rays. 


LIGHT.  413 

Ltjfht.          The  eye  can  judge  pretty  well,  by  practice,  of  the  destruction  of  colour,  and  indistinctness  in  the  edges  of     P»rt  H- 
~-*^-~~'  objects,  when  prisms  are  made  to  act  in  opposition  to  one  another,  as  above  described ;  but  (owing  to  causes  *"- •— v^"' 
presently  to  be  considered)  the  compensation  is  never  perfect,  and  there  always  remains  a  small  fringe  of  uncor-       430. 
reeled  purple  on  one  side,  and  green  on  the  other,  when  the  eye  is  best  satisfied  ;  so  that  observations  of  dispersive 
powers   by  this  method  are  liable  to  a  certain  extent  of  error,   and,   indeed,  precision  in   this   department  of 
optical  science  is  very  difficult  to  obtain. 

To  determine  the  dispersive  power  of  a  medium,  having  formed  it  into  a  prism,  and  measured  by  the  goniometer,       431 
or  otherwise,  its  refracting  angle,  and  ascertained  its  refractive  index,  the  next  step  is  to   find  the  refracting  To  deter- 
angle  of  a  prism  of  some  standard  medium,  which  shall  exactly  compensate  its  dispersion,  so  as  to  produce  m'ne  t'le 
a  refraction  as  nearly  as  possible  free  from    colour.     But  as  it  is    impossible    to  have  a  series    of   standard  ^j.sPersi"n 
prisms  with  every  refracting  angle  which  may  be    requisite,  it  becomes    necessary  to  devise  some  means  of 
varying  the  refracting  angle  of  one  and  the  same  prism  by  insensible    gradations.      Many  contrivances  may 
be  had  recourse  to    for  this.     Thus,  first,    we  may    use  a  prism    composed   of  two  plates  of  parallel  glass,  Prisms  witli 
united  by  a  hinge,  or  otherwise,  and  enclosing  between  them  a  fluid,  which  may  be  prevented  from  escaping  variable  re- 
either  by  capillary  attraction,  if  in  very  small  quantity,  or  by  close-fitting  metallic  cheeks,  forming  a  wedge-  fraclin6 
shaped  vessel,  if  in  larger.     This    contrivance,  however,  is  liable  to  a  thousand  inconveniencies  in  practice.  j"°  -I  j 
Secondly,  we  may  use  two  prisms  of  the  same  kind  of  glass,  one  of  which  has  one  of  its  faces  ground  into 
a  convex,  and  the    other  into  a    concave    cylinder,  of  equal    curvatures,  having  their    axes    parallel    to    the 
refracting  edges.     These  being  applied  to  each  other,    and  one  of  them  being    made  to    revolve  round  the  Another 
common  axes  of  the  two  cylindric  surfaces  upon  the  other,  the  plane  faces  will  evidently  be  inclined  to  each  construction 
other  in    every  possible  angle  within  the  limits  of  the  motion,  (see  fig.   100,   a,   b,  exhibiting  two  varieties  ilg-  1( 
of  this  construction.)     The  idea,  due,  we  believe,  to  Boscovich,  is  ingenious,  but  the  execution  difficult,  and 
liable  to  great  inaccuracies. 

The  following  method  succeeds  perfectly  well,  and  we  have  found  it  very  convenient  in  practice.  Take  a  432. 
prism  of  good  flint  glass,  whose  section  is  a  right  angled  triangle,  ABC,  having  the  angle  A  about  30°  Third  con- 
or  35°,  C  being  the  right  angle,  and  whose  length  is  twice  the  breadth  of  the  side  A  C  ;  and,  having  ground  p™0^"' 
and  polished  the  side  A  C,  and  the  hypothenuse  of  the  prism  to  true  planes,  cut  it  in  half,  so  as  to  form  ]Q2  ' 
two  equal  prisms  with  one  face  in  each  a  square,  and  whose  refracting  angles  (A,  A')  cannot,  of  course, 
be  otherwise  than  exactly  equal.  Cement  the  square  faces  together  very  carefully  with  mastic,  so  that  the 
edges  A,  A',  shall  be  on  opposite  sides  of  the  square  surface,  which  is  common  to  both ;  and  then,  making 
the  whole  solid  to  revolve  round  an  axis  perpendicular  to  the  common  surface,  and  passing  through  its  centre, 
grind  off  all  the  angles  of  the  squares  in  the  lathe,  and  the  whole  will  be  formed  into  a  cylindrical  solid, 
with  oblique,  parallel,  elliptical,  plane  ends,  as  in  fig.  101.  Then  separate  the  prisms,  (by  warming  the 
cement,)  and  set  each  of  them  in  a  separate  brass  mounting,  as  in  fig.  102,  so  as  to  have  their  circular  faces 
in  contact,  and  capable  of  revolving  freely  upon  each  other  about  their  common  centre.  The  lower  one  is  fixed 
in  the  centre  of  the  divided  circle  D  E,  while  the  mounting  of  the  upper  or  moveable  one  carries  an  arm  with  an 
adjustable  vernier  reading  off  to  tenths  of  degrees,  or,  if  necessary,  to  minutes.  The  whole  apparatus  is  set  in  a 
swing  frame  between  plates,  which  grasp  the  divided  plate  by  a  groove  in  its  edge,  allowing  a  motion  in  its  own 
plane,  and  a  capability  of  adjusting  it  to  any  required  position,  so  as  to  admit  of  the  compound  prism  deviating 
an  incident  ray  in  every  possible  plane,  and  under  every  possible  situation,  with  respect  to  the  faces  of  the 
prisms.  It  is  evident,  that  in  the  position  here  represented,  where  the  prisms  oppose  each  other,  (and  at  which 
the  vernier  must  be  set  to  read  off  zero,)  the  refracting  angle  is  rigorously  nothing ;  and  when  turned  round  180°, 
since  the  prisms  then  conspire,  their  combined  angle  must  be  double  that  of  each.  In  intermediate  situations, 
the  angle  between  the  planes  of  their  exterior  faces  must,  of  course,  pass  through  every  intermediate  state,  and 
(by  spherical  trigonometry)  it  is  readily  shown,  that  if  0  be  the  reading  off  of  the  vernier,  or  the  angle  of  rotation 
of  the  prisms  on  each  other  from  the  true  zero,  the  angle  of  the  compound  prism  will  be  had  by  the  equation 

A  0 

sin  -g-    =  sin  —  .  sin  (A)  (b) 

where  (A)  is  the  refracting  angle  of  each  of  the  simple  prisms,  and  A  the  angle  of  the  compound  one. 

To  use  this  instrument,  place  the  prism  A',  whose  dispersive  power  is  to  be  compared  with  the  medium  of  433. 
which  the  standard  prism  (A)  is  formed,  with  its  edge  downwards  and  horizontal,  before  a  window,  and,  selecting  How  us«i, 
one  of  the  horizontal  bars  properly  situated,  fix  it  so  that  the  refraction  of  this  bar  shall  be  a  minimum,  or  till, 
on  slightly  inclining  the  prism  backwards  and  forwards,  the  image  of  the  bar  appears  stationary.  Then  take 
the  standard  compound  prism,  adjust  it  to  zero,  and  set  it  vertically  on  its  frame  behind  the  first  prism.  Move  its 
index  a  few  degrees  from  zero,  and  turn  the  divided  circle  in  its  own  plane,  till  the  refraction  so  produced  by  the 
second  prism  is  contrary  to  that  produced  by  the  first.  The  colour  will  be  found  less  than  before .  continue 
this  till  the  colour  is  nearly  compensated,  then,  by  means  of  the  swing  motion,  and  of  the  motion  round  the 
vertical  axis,  adjust  the  apparatus  so  that  two  of  the  window  bars,  a  horizontal  and  a  vertical  one,  seen  through 
both  prisms,  shall  appear  to  make  a  right  angle  with  each  other,  (an  adjustment,  at  first,  rather  puzzling,  but 
which  a  little  practice  renders  very  easy.)  Then  complete  the  compensation  of  the  colour ;  verify  the  position  of 
the  standard  prism,  (by  the  same  test,)  and  finally  read  off  the  vernier,  and  the  required  angle  A  of  the  com- 
pensating prism  is  easily  calculated  by  the  equation  (6).  This  calculation  may  be  saved  by  tabulating  the  values 
of  A  corresponding  to  those  of  0,  (the  value  of  (A)  being  supposed  known  by  previous  exact  measures,)  or,  by 
graduating  the  divided  circle  at  once,  not  into  equal  parts  ot  0,  but  according  to  such  computed  values  of  A,  so 
as  to  read  off  at  once  the  value  of  the  angle  required. 


414  LIGHT. 

Light.          A  simpler,  perhaps,  on   the  whole,  a  better,  method  of   comparing  the  dispersions  of  two  prisms,  is  one     Part  II. 

v— —\r-~~'  proposed  and  applied  extensively  by  Dr.  Brewster,  in  his  ingenious  Treatise  On  New  Philosophical  Instrument*,  S—-Y— • 
434.       a  work  abounding  with  curious  contrivances  and  happy  adaptations.     It  consists  in  varying,  not  the  refracting 

Another        angle  of  the  standard  prism,  but  the  direction  in  which   its  dispersion  is  performed.     It  is  manifest,  that  if  we 
i   db    can  Pr°duce  from  a  h'ne  of  white  light,  by  means  of  a  standard  prism  any  how  disposed,  a  coloured  fringe,  in 

Dr  Brew-    which  the  colours  occupy  the  same  angular  breadth  as  in   that  produced  by  a  prism  of  unknown   dispersion  ; 

ster  then,  the  latter,  being  made  to  refract  this  fringe  in  a  direction  perpendicular  to  its  breadth,  and  opposite  to  the 

order  of  its  colours,  must  destroy  all  colour  and  produce  a  compensated  refraction  ;  and  therefore  if  the  position 
of  the  standard  prism  which  produces  such  a  fringe  be  known,  the  dispersion  of  the  other  may  be  calculated. 
To  accomplish  this,  let  A  B  be  a  horizontal  luminous  line  of  considerable  length,  and  let  it  be  refracted  downwards, 

Fig.  103.  but  obliquely  in  the  direction  A  a,  B  b,  by  a  standard  prism  whose  dispersion  is  greater  than  that  of  the  prism  to  be 
measured.  Then  it  will  form  an  oblique  spectrum  abb' a',  ab  being  the  red,  and  a'  b'  the  violet ;  and  the  angular 
breadth  of  this  coloured  fringe  will  beam  =  a  a'  X  sin  inclination  of  the  plane  of  refraction  to  the  horizon.  Now, 
let  the  prism  whose  dispersion  is  to  be  measured  be  made  to  refract  this  coloured  band  vertically  upwards ;  then,  if 
the  plane  of  the  first  refraction  be  so  inclined  to  the  horizon  that  the  angle  subtended  by  a  m  at  the  eye  shall  be  just 
equal  to  the  angle  of  dispersion  of  the  other  prism,  all  the  colours  of  the  rectangular  portion  6  ca'  d  will  be  made  to 
coalesce  in  the  horizontal  line  A'  B',  which  will  appear  therefore  free  from  colour,  except  at  its  extremities  A'  B', 
where  the  coloured  triangles  etc  a',  b  d  b'  will  produce  a  red  termination  A' A"  and  a  blue  one  B'B"at  the 
respective  ends  of  the  line  to  which  they  correspond.  Hence,  if,  the  second  prism  remaining  fixed,  with  its  edge 
downwards  and  parallel  to  the  horizon,  the  other  or  standard  prism  be  turned  gradually  round  in  the  plane  perpen- 
dicular to  its  principal  section,  a  position  must  necessarily  be  found  where  the  twice  refracted  line  A'  B '  will  appear 
free  from  colour  both  above  and  below.  In  this  position  let  it  be  arrested,  and  the  angle  of  inclination  of  its  edge 
to  the  horizon  read  off,  its  complement  is  the  angle  aa'm,  which  we  will  call  0.  Let  us  now  suppose  each 
prism  adjusted  to  its  position  of  minimum  deviation,  and  (as  it  is  a  matter  of  indifference  which  is  placed  first) 
let  the  prism  to  be  examined  or  the  fixed  prism  be  placed  next  the  object.*  Then,  D'  and  D  being  the  total 
deviations  produced  by  the  fixed  and  revolving  prisms  on  the  extreme  red  ray,  we  must  have 

A'  A'  +  D'  A  A  +  D 

i  D  —  B  D  .  sin  0  =  o ;         or  6  X .  Bin  — —  .  sec =  S  p  .  sin  -—  .  sec .  sin  S, 

9m  B  IB 

whence  we  obtain 

p'  £  u'         u  —  1  u          /*  —  1        tan  i  (A  +  D) 

__*_          —    w      '  '  —        ' r  ..-_.__.  **      !! -_        Sift  v  '  { C I 

where  the  angles  £  (A  -f  D)  and  £  (A'  +  D')  are  given  by  the  equations 

sin  \  (A  +  D)  =  n  .  sin  x  £  A  ;         sin  £  (A1  +  D')  =  ft' .  sin  \  A' ; 

from  which  formula,  0  being  known,  and  also  the  angles  ana  efractive  indices  of  the  two  prisms,  the  ratio  of 
their  dispersions  is  found. 

435  By  these,  or  other  similar  methods,  may  the  dispersions  of  any  media  be  compared  with   those  of  any  other 

Absolute      taken  as  a  standard.     If  the  media  be  solid,  they  must  be  formed  into  prisms  ;  if  fluid,  they  must  be  enclosed  in 

dispersive     hollow  prisms  of  truly  parallel  plates  of  glass,  whose  angles  must  be  accurately  determined,  (and  one  of  which 

powers,how  wj|j  serve  for  any  nuraber  of  fluids.)     But  to  ascertain  directly  the  dispersion  of  that  standard  prism,  we  must 

lsttaByeniea-  Pursue  a  different  course.     The  first  method  which  obviously  presents  itself,  is  to  measure  the  actual  length  of 

suring  the     the  solar  spectrum  cast  by  a  prism  of  given  refracting  angle  ;  but  the  light  of  the  spectrum  dies  away  so  inde 

spectrum  on  finitely  at  both  ends,  and  its  visible  extent  varies  so  enormously  with  the  brightness  of  the  sun,  and  the  more 

a  screen.      or  ]ess  perfect  exclusion  of  extraneous  light,  that  nothing  certain  can  be  concluded  from  such  measures.     Yet, 

if  the  brighter  rays  of  the  spectrum  be  destroyed,  and  the  eye  defended  from  all  offensive  light  by  a  glass  which 

permits  only  the  extreme  red  and  violet  rays  to  pass,  (see  Index,  Absorption,)  some  degree  of  accuracy  may  be 

obtained  by  this  means.     A  method  founded  on  this  principle  has  been  described  by  the  writer  of  these  pages 

Fig.  104.     in  the  Transactions  of  the  Royal  Society  of  Edinburgh,  vol.  ix.  as  follows :  Let  A  and  B  be  two  vertical  rect- 

Snd.Another  angular  slits  in  a  screen  placed  before  an  open  window,  the  one  being  half  the  length  of  the  other,  and  at  a 

method.        known  distance  from  each  other.     The  eye  being  guarded  as  above  described,  let  the  slits  be  refracted  by  the 

prism  (in  its  minimum  position)  from   the  longer  towards  the  shorter.     Then  will  a  red  and  violet  image  of 

each  a,  b,  and  a',  b'  be  seen.     Now  let  the  prism  be  removed  from  the  slits,  (or  vice  versa,)  still  preserving  its 

position  of  minimum  deviation,  till  the  violet   image  of  the  longer  slit  exactly  falls  upon  and  covers  the  red 

image  of  the  shorter,  as  in  the  position  a  b  of  the  figure.     Then  it  is  obvious,  that  the  distance  between  the 

slits,  divided  by  their  distance  from  the  prism,  is  the  sine  of  the  total  angle  of  dispersion,  or  is  equal  to  S  D, 

and  this  being  known 


and 

_    S  D       cos  J  (A 

L   +   D) 

J"~        2                sin: 

^A 

•  Dr.  Brewster  has  chosen  a  somewhat  different  position,  (Treatiie,  fyc.  p.  296,)  with  a  view  to  simplify  the  formulae ;  but  it  does  not 
•ppear  to  us  that  any  advantage  is  gained  in  that  respect  by  his  arrangement. 


L  I  G  H  T. 


415 


Light.  But  all  these  methods  are  only  rude  approximations,  as  the  great  discrepancies  of  the  results  hitherto  obtained 
^-V^^  by  them  abundantly  prove  ;  thus,  the  dispersions  of  various  specimens  of  flint  glass,  obtained  by  the  method  last 
described,  come  out  no  less  than  one-sixth  larger  than  those  previously  given  by  Dr.  Brewster.  The  only  method 
which  can  really  be  relied  on  is  that  practised  by  Fraunhofer,  (where  the  media  can  be  procured  in  a  state  of  suffi- 
cient purity  and  quantity  for  its  application  ;)  and  consists  in  determining,  with  astronomical  precision,  by  direct 
measures,  the  values  of  ft  for  the  several  points  of  definite  refrangibility  in  the  spectrum,  marked,  either  by  the 
fixed  lines,  or  by  the  phenomena  of  coloured  flames  or  absorbent  media.  (See  Index,  Flames — Absorption.) 
By  taking  advantage  of  the  properties  of  the  latter,  a  red  ray,  of  a  refrangibility  strictly  definite,  may  be 
insulated  with  great  facility ;  and  as  it  lies  so  near  the  extremity  of  the  spectrum  as  not  to  be  perceptible  till  all 
the  brighter  rays  are  extinguished,  it  is  invaluable  as  a  fixed  term  in  optical  researches,  and  will  always  be  un- 
derstood by  us  in  future,  when  speaking  of  the  commencement  of  the  spectrum,  or  the  extreme  red,  even  though 
a  red  ray  still  less  refrangible  should  be  capable  of  being  discerned  by  careful  management,  and  in  favourable 
circumstances.  In  like  manner,  by  the  simple  artifice  of  putting  a  little  salt  into  a  flame,  a  yellow  ray  of  a 
character  perfectly  definite  is  obtained,  which,  it  is  very  remarkable,  occupies  precisely  the  place  in  the  scale  of 
refrangibility  where  in  the  solar  spectrum  the  dark  line  D  occurs,  (Art.  418,  419.)  These,  and  the  fixed  lines 
there  mentioned,  leave  us  at  no  loss  for  rays  identifiable  at  all  times  and  in  all  circumstances,  (with  a  good  appa- 
ratus,) and  enable  us  to  place  the  doctrine  of  refractive  and  dispersive  powers  on  the  footing  of  the  most  accu- 
rate branches  of  science. 

The  following  table,  extracted  from  Fraunhofer' s  Essay  on  the  Determination  of  Refractive  and  Dispersive 
Powers,  $c.  contains  the  absolute  values  of  the  index  of  refraction  ft  for  the  several  rays  whose  places  in  the 
spectrum  correspond  to  the  seven  lines  B,  C,  D,  E,  F,  G,  H,  assumed  by  him  as  standards  (see  Art.  419,  &c.) 
for  several  different  specimens  of  glass  of  his  own  manufacture,  and  for  certain  liquids.  These  values,  for  dis- 
tinction's sake,  we  may  designate  by  the  signs  /t  (B),  ft  (C),  p,  (D),  &c. 


Part  II. 

436. 

Method 
employed  by 
Fraunhofer. 


Use  of  the 
fixed  lines. 


437 


Table  of  the  refractive  indices  of  various  glasses  and  liquids  for  seven  standard  rays. 


Specific 

Values  of 

A 

gravity. 

A-(B) 

/.(C) 

/»(D) 

ME) 

/•(F) 

MG) 

JT(i) 

Flint  glass,  No.  13  

3.723 

1  627749 

1  629681 

1  635036 

1  642024 

1  648260 

1  660285 

1  67]  06* 

Crown  glass  No.  9   . 

2.535 

1  525832 

1  526849 

1  529587 

1  533005 

1  536052 

1  541657 

1   IS4firififi 

Water  

1.000 

1  330935 

1  331712 

1  333577 

1  335851 

1  337818 

1  341293 

1  344177 

Water,  another  experiment 

1.000 

1.330977 

1.331709 

1.333577 

1.335849 

1.337788 

1.341261 

1.344162 

Solution  of  potash    .    ... 

1.416 

1  399629 

1  400515 

1  402805 

1  405632 

1  408089 

1  412579 

1  416368 

Oil  of  turpentine  

0.885 

1  470496 

1  471530 

1  474434 

1  478353 

1  481736 

1  488198 

1  493874 

Flint  glass,  No.  3    

3.512 

1.602042 

1.603800 

1.608494 

1.614532 

1.620042 

1.630772 

1.640373 

Flint  "-lass,  No.  30  

3.695 

1  623570 

1  625477 

1  630585 

1  637356 

1  643466 

1  655406 

1  666072 

Crown  glass,  No.  13  .... 

2.535 

1.524312 

1.525299 

1.527982 

1.531372 

1.534337 

1.539908 

1.544684 

Crown  glass,  letter  M.  .  .  . 

2.756 

1.554774 

1.555933 

1.559075 

1.563150 

1.566741 

1.573535 

1.579470 

Flint  glass,  No.  23  ....  \ 
Prism  of  60°  15'  42''  j 

3.724 

1.626596 

1.628469 

1.633667 

1.640495 

1.646756 

1.658848 

1.669686 

Flint  glass,  No.  23  \ 
Prism  of  45°  23'14"j 

3.724 

1.626564 

1.628451 

1.633666 

1.640544 

1.646780 

1.658849 

1.669680 

The  above  table  renders  very  evident  a  circumstance  which  has  long  been  recognised  by  experimental  opticians,       438. 
and  which  is  of  great  importance  in  the  construction  of  telescopes,  viz.  the  irrationality,  (as  it  has  been  termed,)  or  Identifica. 
want  of  proportionality  of  the  spaces  occupied  in  spectra  formed  by  different  media  by  the  several  coloured  rays,  tion  of  a  ra> 
or  by  those  whose  refrangibilities,  by  any  one  standard  medium,,  lie  between  given  limits.      If  we  fix   upon  ^ its  place 
water,  for  example,  as  a  standard  medium,  (and  we  see  no  reason  why  it  should  not  be  generally  adopted  as  a  sp^nim"' 
term  of  reference  in  this,  as  in  other  physical  inquiries— of  course  at  a  given  temperature— that  of  its  maximum 
density,  for  instance,)  it  is  obvious,  that  any  ray  may  be  identified  by  stating  its  index  of  refrangibility  by  water  ; 
thus,  a  scale  of  refrangibilities,  which,  for  brevity,  we  shall  term  the  water  scale,  is  established ;  and  so  soon  as  we 
know  the  refractive  index  of  a  ray  from  vacuum  into  water,  we  have  its  place  in  the  water  spectrum,  its  colour, 


416  LIGHT. 

Light      and  its  other  physical  properties  (so  far  as  they  depend  on  the  refrangibility  of  the  ray)  determined.     Thus     Part  II. 
v-p"v~»/'  1.333577  being  known  to  be  the  refractive  index  for  a  ray  in  water,  that  ray  can  be  no  other  than  the  particular  v—  •  -v~—  i 
ray  D,  whose  colour  is  pale  orange-yellow,  and  which  is  totally  deficient  in  solar  light,  and  peculiarly  abundant 
in  the  light  of  certain  flames.     Now  let  x  be  the  refractive  index  of  any  ray  whatever  for  water,  or  its  place  in 
the  water  scale.     Then  it  is  evident,  that  its  refractive  index  for  any  other  medium  must  of  necessity  be  a  function 
of  x,  because  the  value  of  x  determines  this  and  all  the  other  properties   of  the  ray.     Hence  we  must  have 
between   /i    and  x  some    equation  which  may   be   generally  represented   by  p.  =  F  (x)  ;  F  (x)  denoting   a 
function  of  x. 

i39.          To  determine  the  form  of  this  function,  we  must  consider,  that  if  A  be  the  very  small  angle  of  a  prism, 

Function  of  A  A  +  D 

refrangibi-    alMJ  D  the  deviation  produced  by  it  at  the  minimum,  we  have  /*.  —  -=     —  -  -  ,  or  D  =  (/*  —  1)  A.     Hence, 

lity.  *  " 

supposing  A  the  refracting  angle  constant,  the  deviation  is  proportional  to  fi  —  1.  Now,  since  in  all  media,  as 
well  as  in  water,  the  deviations  observe,  at  least,  the  same  order,  being  always  least  for  the  red  and  greatest  for 
the  violet,  it  follows,  that  in  all  media  p.  —  1  increases  as  x  increases  ;  so  that,  supposing  x0  to  be  the  index  of 
refraction  in  the  water  scale  for  the  first  visible  red  ray,  or  the  commencing  value  of  x,  and  /«0  the  index  for  the 
same  ray  in  the  other  medium,  (JJL  —  1)  —  (/»0—  1),  or  ft  —  ft0  must  increase  with  ^  —  x0;  and  since  they 
vanish  together,  we  may  represent  the  one  in  a  series  with  indeterminate  coefficients,  and  powers  of  the  other, 
thus 

/.  -  fi0  =  A  (x  -  x0)  +  B  (x  -  ,r0)«  +  C  (x  -  x0)  3  +  &c.  ; 

or,  which  comes  to  the  same  thing,  a  b,  c,  &c.,  representing  other  indeterminate  coefficients,  (j;0  —  1  being 
constant,) 

6  .     f  Ii'  +  &,  W 


. 
u0  —  1  \x0  —  I 

440.  The  simplest  hypothesis  we  can  form  respecting  the  values  of  a,  b,  &c.  is  that  which  makes  a  =  1,  and  b, 

Hypothesis 

of  constant  .     ,,                                                                    _,,.       .           /*  —  /*o              J        To 

dispersion  ant"  a''  tne  other  coefficients  vanish.     This  gives    —  —  —  -  =   -  —  -  . 

in  all  media.  ^°~  *°  ~ 

We  have  before  used  £  p.  to  denote  what  is  here  signified  by  /*  —  /»„,  viz.  the  difference  between  the  refractive 

&  p 
indices  of  any  ray  in  the  spectrum,  and  that  at  its  commencement  ;  and  we  have  denoted  by  --  —  the  same 

quantity  which  is  here  expressed  by  —  --  ^-.     This  then  is  the  expression,  in  our  present  notation,  of  the 

Nut  the  law  dispersive  power  of  the  medium  ;  and  the  equation  now  under  consideration  therefore  indicates,  that,  on  the 
uf  nature,     hypothesis  made,  the  dispersive  power  of  the  medium  must  necessarily  be  the  same  with  that  of  water;    and 

of  course  (supposing  this  hypothesis  to  be  founded  in  the  nature  of  light)  all  media  must  have  the  same  dis- 

persive power.     This,  as  we  have  already  seen,  is  not  the  case. 

Nor  that  of     The  next  simplest  hypothesis  is  that  which  admits  a  as  an  arbitrary  constant  determined  by  the  nature  of  the 
proportional  medium,  but  still  makes  b,  c,  &c.  =  o.     This  reduces  the  equation  to 

dispersions. 

/»  ~  ^o    _  a        x  —  xo  . 

ft,  -  !  *o  ~  !   ' 

consequently  (if  /i'  and  x'  be  any  other  corresponding  values  of  /»  and  x)  we  must  have  also 


441 


Hence,  if  this  hypothesis  be  correct,  and  /.,  x  and  /,  x'  be  any  two  pairs  of  corresponding  refractive  indices 

for  rays  however  situated,  the  fraction  ^  ~  ^  must  be  invariable.     The  foregoing  table,  however,  shows  very 

x   —  x 

distinctly  that  this  is  far  from  being  the  case.     Thus,  if  we  take  the  flint  glass,  No.  13,  the  comparison  of  the 
two  rays  B  and  C  gives  for  the  value  of  the  fraction  in  question  2.562  ;  and  if  we  compare  in  hke  manner  t 
rays  C  and  D,  D  and  E,  E  and  F,  F  and  G,  G  and  H  respectively,  we  obtain  the  values  2.871,  3 
3.460,  3.726  ;  the  great  deviation  of  which  from  equality,  and  their  regular  progression,  leaves  no  d, 
incompatibility  of  the  hypothesis  in  question,  as  a  general  law,  with  nature.     If  we  institute  the  same  comparison 
for  the  other  media  in  the  table,  we  shall  find  the  greatest  diversity  prevail ;  and  if,  instead  of  water,  we  assume 
any  other  as  a  standard,  the  same  incompatibility  will  be  found.     Thus  if  the  flint  glass ,  No   ! 13,  be  compared 
with  oil  of  turpentine,  we  find  for  the  values  of  the  series  of  fractions  in  question,  1.868,  1.844,  1.7SJ,  1.S4J, 
1.861,   1.899,  which  first  diminish  to  a  minimum  and  then  increase  again,  &c. 

It  follows  from  this,  that  the  proportion  which  the  several  coloured  spaces  (or  the  intervals  V,  1J 

&c.)  bear  to  each  other  in  spectra  formed  by  different  media,  is  not  the  same  in  all.     Thus  taking  the  green 
ray  E  for  the  middle  colour,  and  calling  all  that  part  of  the  spectrum  which  lies  on  the  red 


LIGHT. 


417 


Light,      and  all  on  the  other  side  the  blue  portions,  the  ratio  of  the  spaces  occupied  by  the  red  and  blue  in  any  spectrum 

*~V~""  will  be  represented  by  the  fraction  ,  \.  •      Now  tne  values  of  this  in  the  several  media  of  the 

p  (E)  —  p  (B) 

foregoing  table  are  set  down  in  the  following  list  : 


PartlL 


Flint   No.  23 

2  0922 

Crown,  M  

1.9484 

Flint,  No.  30.  . 

2.0830 

Crown,  No.  9   .... 

1.8905 

Flint,  No.  3  

2.0689 

Crown,  No.  13.  ... 

1.8855 

Flint,  No.  13  

20342 

Solution  of  potash. 

1.7884 

Oil  of  turpentine   .  . 

1.9754 

Water  

1.6936 

Incommen- 
surability of 
the  coloured 
spaces  in 
.  pectra  of 
different 
media. 


442. 


Here  we  see  that  the  same  coloured  spaces  which  in  the  flint  No.  23  are  in  the  ratio  of  21  :  10,  in  the  water 
spectrum  are  only  in  the  ratio  of  17  :  10  (nearly,)  so  that  the  blue  portion  of  the  spectrum  is  considerably  more 
extended  in  proportion  to  the  red  in  the  flint  glass  than  in  the  water  spectrum. 

From  this  it  follows,  that  if  two  prisms  be  formed  of  different  media  (such  as  flint  glass  and  water)  of  such 
refracting  angles  as  to  give  spectra  of  equal  total  lengths,  and  these  be  made  to  refract  in  opposition  to  each  Secondary 
other,  although  the  red  and  violet  rays  will,  of  course,  be  united  in  the  emergent  beam,  yet  the  intermediate  sPectra- 
rays  will  still  be  somewhat  dispersed,  the  water  prism  refracting  the  green,  or  middle  rays  more  than  in  pro- 
portion to  the  extremes ;  consequently,  if  a  white  luminous  line  be  the  object  examined  through  such  a  combi- 
nation, instead  of  being  seen  after  refraction  colourless,  it  will  form  a  coloured  spectrum  of  small  breadth 
compared  with  what  either  prism  separately  would  form,  and  having  one  side  of  a  purple  and  the  other  of  a 
green  tint.  Any  dark  object  viewed  against  the  sky  (as  a  window  bar)  will  be  seen  fringed  with  purple  and 
green  borders,  the  green  lying  on  the  same  side  of  the  bar  with  the  vertex  of  the  flint  prism  ;  because  in  such 
a  combination,  green  must  be  considered  as  the  most,  and  purple  as  the  least,  refrangible  tint ;  and  the  flint 
prism,  of  necessity,  having  the  least  refraction  in  this  case,  the  most  refrangible  fringe  will  lie  towards  its  vertex, 
that  being  the  least  refracted  side  of  the  bar ;  for  the  same  reason  that,  when  seen  through  a  single  prism,  a 
dark  object  on  a  white  ground  appears  fringed  with  blue  on  its  least  refracted  edge.  (Art.  429.) 

This  result  accords  perfectly  with  observation.  Clairaut,  and,  after  him,  Boscovich,  Dr.  Blair,  and  Dr 
Brewster,  have  severally  drawn  the  attention  of  opticians  to  these  coloured  fringes,  or,  as  they  may  be  termed, 
secondary  spectra,  and  demonstrated  their  existence  in  the  most  satisfactory  manner.  Dr.  Brewster,  in  parti- 
cular, has  entered  into  a  very  extensive  and  highly  valuable  series  of  experiments,  described  in  his  Treatise  on 
new  philosophical  instruments,  and  in  his  paper  on  the  subject  in  the  Edinburgh  Transactions ;  from  which  it 
follows,  that  when  a  compound  prism,  consisting  of  any  of  the  media  in  the  following  list  refracting  in  oppo- 
sition to  each  other,  unites  the  red  and  violet  rays,  the  green  will  be  deviated  from  their  united  course  by 
the  combination,  in  the  direction  of  the  refraction  of  that  medium  which  stands  before  the  other  in  order : 


443. 


1.  SULPHURIC  ACID. 
2.  Phosphoric  acid. 
3.  Sulphurous  acid. 
4.  Phosphorous  acid. 
5.  Super-sulphuretted  hydrogen. 
6.  WATER. 
7.  Ice. 
8.  White  of  egg. 
9.  Rock  crystal. 
10.  Nitric  acid.  ^ 
11.  Prussic  acid. 
12.  Muriatic  acid. 
13.  Nitrous  acid. 
14.  Acetic  acid. 
15.  Malic  acid. 
16.  Citric  acid. 
17.  Fluor  spar. 
18.  Topaz,  (blue.) 
19.  Beryl. 
20.  Selenite. 
21.  Leucite. 
22.  Tourmaline. 
23.  Borax. 
24.  Borax,  (glass  of.) 
25.  Ether. 
26.  Alcohol. 
27.  Gum  Arabic. 
28.  CROWN  GLASS. 
29.  Oil  of  almonds. 
30  Tartrate  of  potasti  and  soda. 

31.  Gum  juniper. 
32.  Rock  salt. 
33.  Calcareous  spar. 
34.  Oil  of  ambergris. 

61.  Oil  of  nutmeg's. 

64.  Amber. 
65.  Oil  of  spearmint. 

o  /  .            p°ppy- 

39.  Zircon. 
40.  FLINT  GLASS. 
41.  Oil  of  rhodium. 
42.  .  rosemary. 

71.  Canada  balsam. 
72.  Oil  of  lavender. 
73.  Muriate  of  antimony. 
74.  Oil  of  cloves. 
75.  sweet  fennel  seeds. 
76.  Red-coloured  glass. 
77.  Orange-coloured  glass. 
78.  Opal-coloured  glass. 
79.  Acetate  of  lead,  (melted.) 
80.  Oil  of  amber. 
81.  •              sassafras. 

44.  Balsam  of  capivi. 
45.  Nut  oil. 
46.  Oil  of  savine.    . 

49.  Nitrate  of  potash. 
50.  Diamond. 
51.  Resin. 
52.  Gum  copal. 
53.  Castor  oil. 
54.  Oil  of  chamomyle. 

83.  •  anise  seeds. 
84.  Essential  oil  of  bitter  almonds. 
85.  Carbonate  of  lead. 
86.  Balsam  of  Tolu. 
87.  Sulphuret  of  carbon. 
86.  Sulphur. 
89  Oil  of  cassia. 

57.  marjoram. 

Dr.  Brew- 

ster's  table 
of  media 
according 
to  action  on 
green  light. 


VOL.  IV. 


3  i 


418 


LIGHT. 


It  is  evident  from  this  table,  that  (generally  speaking)  the  more  refractive  a  medium  is,  the  greater  is  the     Part  II. 
extent  of  the  blue  portion  of  its  spectrum  compared  with  the  red. 

If  two  prisms  of  the  proper  refracting1  angles,  composed  of  media  not  very  remote  from  each  other  in 
this  list,  be  made  to  oppose  each  other,  the  secondary  spectrum  will  be  small,  and  the  refraction  almost  perfectly 
colourless.  Such  a  combination  is  said  to  be  achromatic,  (a-x/jtyta.) 

The  existence  of  the  secondary  spectrum,  while  it  renders  the  attainment  of  perfect  achromaticity  impossible, 
by  the  use  of  two  media  only,  shows,  also,  that  in  a  theoretical  point  of  view  we  are  not  entitled  to  neglect  the 
coefficients  6,  c,  &c.  of  the  equation  (<2),  Art.  439.  The  law  of  nature  probably  requires  the  series  to  be  continued 
to  infinity  ;  and  if,  by  way  of  uniting  three  rays,  we  employ  prisms  of  three  media,  tertiary  spectra,  and  after 
them  still  others  in  succession,  would  doubtless  be  found  to  arise.  These,  however,  will  be  small  in  comparison 
of  each  other. 

The  table  (Art.  437)  gives  us  the  means  of   computing  the   coefficients  on  which  they  depend  for  the 


Light. 

^— ^— ' 

444. 

445. 

Achromatic 
refraction. 

446. 

Dispersiv« 
powers  of 
higher 
orders. 
Tertiary 
spectra. 
447. 
Computa- 
tion of  their  particular  media   there   stated.     If    we   put 
coefficients. 


=  P,  and 


=  p,   and   suppose  P,  P',  P'1, 


p,  p',  p'1,  &c.  to  be  the  values  of  P  and  p  corresponding  to  the  several  values  of  p,  and  *  set  down  in  the  table, 
we  shall  have,  for  determining  a,  b,  c,  &c.  in  any  one  of  those  media,  the  equations 


448. 
General 
conditions 
of  achro- 
maticity. 


&c.     P'= 


cp 


&c.     F'  = 


&c. 


and  as  many  such  equations   must  be  used  as  there  are  coefficients  to  determine.     Confining   ourselves   at 
present  to  two,  we  find  P  =  a  p  +  b  p*  ;     P'  =  a  p'  +  b  p"1,  whence 


b=   - 


Pp'-P'ff 


pp'(p'-p)  PP'(P'-P) 

and,  since  it  is  desirable  to  select  rays  as  far  removed  from  each  other  in  the  spectrum  as  possible,  we  shall 
take  fi0  and  x0  from  the  column  /t  (B)  ;  and  determine  P  and  p  by  the  values  in  the  column  /t  (E),  and  F,  p1 
by  those  under  /*  (H).  The  results  will  be  as  follows  : 


Refracting  media. 

Dispersive  powers 
of  the  first  order, 
that  of  water  being 
1.000. 

Dispersive  powers 
of  the  second  order, 
that  of  water  being 
0.000. 

Flint  glass,  No.  13  .. 
Crown  glass.  No.  9  .  . 
Water  

a=  +  1.42580 
0.88419 
1.00000 

b  =  +  7.57705 
2.34915 
0.00000 

Solution  of  potash  .  . 
Oil  of  turpentine  .... 
Flint  glass,  No.  3  
Flint  glass,  No.  30  .  . 
Crown  glass,  No.  13.  . 
Crown  glass,  letter  M  . 
Flint  glass.  No.  23  .  . 

0.99626 
1.06149 
1.29013 
1.37026 
0.87374 
0.90131 
1.37578 

1.13262 
4.58639 
7.63048 
8.44095 
2.49199 
3.49000 
8.66904 

Problem.  To  determine  the  analytical  relation  which  must  hold  good  in  order  that  two  prisms  may  form  an 
achromatic  combination  ;    that  is,  may  refract  a  white  ray  without  separating  the  extreme  colours. 

Resuming  the  equations  and  notation  of  Art.  215,  since  the  prisms  are  placed  in  vacuo,  we  have  to  substi- 

tute u,  —r,  u'  and    —7-  for  u,  u.',  u",  p.'",  in  those  equations  respectively,  and  we  shall  have 
u!  /»' 


fi  .  sin  p  =  sin  a     ~\ 
a'=I  +  P  > 

sin  />'  =  /».  sin  o'  } 


sin  a 


sin  a"  =  /*' 


D  =  a  +  I  +  I'  +  I"  -  / 


and  »    —  I  +  P  ; 

Now,  since  by  hypothesis  the  incident  and  emergent  rays  are  both  colourless,  we  must  have  t  a  =  0,  and 
S  D  =  0,  that  is  S  p"1  —  0,  the  sign  $  being  supposed  to  refer  to  the  variation  of  the  place  of  the  ray  in  the 
spectrum.  Hence  the  two  systems  of  equations  (1)  and  (2)  are  exactly  similar,  in  their  form;  the  former  a* 
relates  to  p,  a,  a.',  p',  and  the  latter  as  to  a'",  p'",  p',  a.".  Now,  the  first  system  gives 

8  ft .  sin  p  -f-  /*  &  p  .  COS  p  =  0  ;          8  at  =  8  p;          $  p'  cos  />'  =  S  ft .  sin  a'  +  /t  &  a'.  COS  J } 

whence,  by  elimination  and  reduction,  we  find 

cm  r 

«/.;  (f) 


COS  p  .  COS  f 


LIGHT. 

Light,      and,  consequently,  by  reason  of  the  analogy  of  the  two  systems  of  equations  pointed  out  above, 


But,  since  a!'  =  1'  +  /,  we  have  &  /=  8  a",  so  that  we  finally  get 

cos  p  .  cos  p'  sin  I         3  fi 

cos  a"',  cos  a"   =        '  "sin  I""  '   Tff  ' 

The  property  expressed  by  this  equation  may  be  thus  stated.  Conceive  the  ray  to  pass  both  ways  outwards 
from  a  point  in  its  course  between  the  two  prisms  ;  then,  in  order  that  the  combination  may  be  achromatic, 
the  product?  of  the  cosines  of  its  incidences  on  the  surfaces  of  each  prism  must  be  to  each  other  in  the  ratio  com- 
pounded of  that  of  the  sines  of  their  respective  refracting  angles,  and  the  differences  of  their  refractive  indices 
for  red  and  violet  rays  ;  besides  which,  they  must  refract  in  opposition  to  each  other,  or  I  and  I"  their  refracting 
angles  must  have  opposite  signs. 

The  combination  of  this  equation  with  the  system  of  equations   above  stated,  expressing  the  conditions  of       449. 
refraction  by  the  prism,  and  their  relative  position  with  regard  to  each  other  (which  is  included  in  the  equation  Progress  of 
a"  =  I'  +  />')  suffice,  algebraically  speaking,  to-  resolve  every  problem  which  can   occur,  of  this  kind  ;  but  the 
final  equations  are  for  the  most  part  too  involved  to  allow  of  direct  solution.     Nevertheless,  the  results  we 
have  arrived  at  will  furnish  occasion  for  remarks  of  moment  ;  and,  first,  since  p'  is  the  angle  of  refraction  from 
the  second  surface  of  the  first  prism,  5  p'  is  the  angular  breadth  of  the  spectrum  produced  by  it  ;  this  is,  there- 
fore, proportional,  cteteris  paribus,  to  the  product  of  the  secants  of  the  angles  of  refraction  at  its  two  surfaces. 
Let  us  trace  the  progress  of  the  variation  of  this,   as  the  incident  ray  changes   its  inclination  to  the  first 
surface,  beginning  with  the  case  when  it  just  grazes  the  surface  from  the  back  towards  the  edge.     In  this  case 

a  =  90°,  sin  p  =  —  ,   consequently  p,  and  therefore  I  +  p  or  a!,  and  therefore  p'  are  all  finite,  and  at  their 

maximum.     Hence  cos  p  .  cos  p'  is  finite,  and  at  its  minimum  ;  and  therefore  S  ^',  or  the  breadth  of  the  spectrum, 

is  also  finite,  but  a  maximum.     As  the  incident  ray  becomes  more  inclined  to  the  surface  p,  and  therefore  a!  and 

5'  diminish,   and  the  denominator  of    5  p'  increases,    so    that    the  breadth    of   the  spectrum   diminishes,   and 

reaches  a  minimum  when  cos  p  .  cos  />'  attains  its  maximum  ;  that  is,  when  d  f  .  tan  p  +  dp'  .  tan  p'  =  0.     Now  Position  of 

this  equation,  substituting  and  reducing  gives,  for  determining  the  value  of  p,  and  therefore  of  a,  or  the  inci-  least  disper- 

dence  when  the  spectrum  is  a  minimum,  s'on  deter- 

mmed. 

/t2  .  sin  (I  +  p)  .  cos  (I  +  2  p)  +  sin  p  =  0.  (h) 

Hence  we  see  that  the  position  which  gives  a  minimum  of  breadth  to  the  spectrum  is  very  different  from  that 
which  gives  a  minimum  of  deviation,  being  given  by  the  above  equation,  which  is  easily  resolved  by  a  table  of 

logarithms,  and  which   shows  at  once  that  p  must  be  greater  than  45°  —  —  . 

After  attaining  the  position  so  determined,  the  breadth  of  the  spectrum  again  increases,  and  continues  to  do 
so  till  the  rays  can  be  no  longer  transmitted  through  the  prism.  At  this  limit  the  emergent  ray  just  grazes  the 
posterior  face  of  the  prism  from  its  thinner  towards  its  thicker  part  g'  ±=  90°,  cos  p1  =  0.  At  this  limit,  therefore, 
the  dispersion  becomes  infinite.  All  these  stages  are  easily  traced  by  turning  a  prism  round  its  edge  between 
the  eye  and  a  candle  ;  or,  better,  between  the  eye  and  the  narrow  slit  between  two  nearly  closed  window-shutters. 

Hence,  as  the  incident  ray  varies  from  the  position  S  E  (fig.  105)  to   S'  E,  and  therefore  the  refracted  from       ^ 
F  G  to  F'  G',  the  breadth  of  the  spectrum  commences  at  a  maximum,  but  finite  value,  diminishes  to  a  minimum  Of  Sst°^!t°"m 
and  then  increases  to  infinity.     The  distribution  of  the  colours  in  the  spectrum,  or  the  breadths  of  the  several  at  extreme 
coloured  spaces  in  any  state  of  the  data,  will  moreover  differ  according  to  the  values  of  p,  p1  and  sin  I;  for  the  incidences. 
equation  (e),  by  assigning  to  5  (i  the  values  which  correspond  in  succession  to  the  intervals  between   red  and  Fi<>-  105. 
orange,  orange  and  yellow,  yellow  and  .green,  &c.  will  give  the  corresponding  values  of  5  p',  or  the  apparent 
breadths  of  these  spaces.     Now  the  denominator  cos  p  .  cos  p'  is  an  implicit  function  of  ft,  and  therefore  varies 
when  the  initial  ray  is  taken  in  different  parts  of  the  spectrum.     The  variation  is  trifling  when  the  angles  p,  p1 
are  considerable  ;  but  near  the  limit,  when  the  ray  can  barely  be  transmitted,  it  becomes  very  great,  the  spectrum 
is  violently  distorted,  and  the  violet  extremity  greatly  lengthened  in  proportion  to  the  red.     The  effect  is  the 
same  as  if  the  nature  of   the  medium  changed  and  descended  lower  in  the  order  of  substances  in  the  table 
Art.  443. 

From  what  has  just  been  said,  we  see  the  possibility  of  achromatising  any  prism,  however  large  its  refracting  451. 
angle,  by  any  other  of  the  same  medium,  however  small  may  be  its  angle  ;  for  since,  by  properly  presenting  a  Achromatic 
prism  to  the  incident  ray,  its  dispersion  may  be  increased  to  infinity  ;  if  made  to  refract  in  opposition  to  another  c.omb'"a- 
whose  dispersion  has  any  magnitude,  however  great,  it  may  be  made  to  counteract,  or  even  overcome  it.  Thus  medium' 
in  fig.  106  the  dispersion  of  the  second  prism  a,  of  small  refracting  angle,  being  increased  by  the  effect  of  its  f\s.  ioe. 
inclined  position,  is  rendered  equal  and  opposite  to  that  of  the  prism  A,  whose  refracting  angle  is  large. 

When  the  prisms  differ  greatly  in  their  angles,  however,  the  second  must  be  very  much  inclined,  so  as  to       453. 
bring  it  near  to  the  limit  of  transmission.     In  this  case,  its  law  of  dispersion,  as  just  shown,  will   be  greatly  Subordinate 
disturbed,  and  rendered  totally  different  from  what  obtains  in  the  other  prisms  ;   so  that  perfect  achromaticity  spectra. 

3  i2 


420  LIGHT. 

Light.      cannot  be  produced  ;  but  when  the  extreme  red  and  violet  rays  are  united,  the  green  will  be  too  little  refracted  by    Part  II. 
1  the  second  prism,  and  a  purple   and  green  spectrum  will  arise,  as  in  the  case  of  prisms  of  different  media.     To  ^— - v^« 
this  spectrum  Dr.  Brewster  (who  was  the  first  to  place  it  in  evidence)  has  given  the  name  of  a  tertiary  spectrum; 
but  it  appears  to  us,  that  this  term  had  better  be  reserved  for  the  spectra  mentioned  in  Art.  446,  and  those  now 
in  question  may  be  called  subordinate  spectra. 

If  a  small  rectangular  object  be  viewed  through  such  a  combination  as  above  described,  in  which  the  prism  A 
is  placed  in  its  position  of  minimum  deviation,  and  achromatised  by  a  second  a,  whose  angle  is  less  than  that 
of  A,  but  not  so  small  as  to  introduce  this  cause  of  colour,  it  will  appear  distorted  in  figure  ;  for  the  sides 
parallel  to  the  edges  of  the  prisms  will  undergo  no  change  in  their  apparent  length,  while  the  breadth  of  the 
rectangle  will  appear  magnified.  For  the  first  prism,  by  reason  of  its  position,  does  not  alter  the  angular 
dimensions  of  objects  seen  through  it;  but  the  second  changes  their  angular  breadth  in  the  ratio  of  dp"'  to 


cos  a 


d  a",  that  is  (bv  differentiation)  in  the  ratio  of  —  —  -  —  :  —  r  to  unity,  a  ratio  which    increases  rapidly  as  the 

COS  p  .  COS  />' 

inclination  of  the  prism  increases,  and  /  approaches  a  right  angle. 

453.  M.  Amici  has  taken  advantage  of  these  properties  to  construct  a  species  of  achromatic  telescope,  which,  at 

Amici's  fjrst  sight,  appears  very  paradoxical,  being  composed  merely  of  four  prisms  of  the  same  kind  of  glass,  with 
plane  surfaces.  To  understand  its  construction,  conceive  a  small  square  object  op  placed  with  the  side  n  parallel 
to  the  refracting  edges  of  a  pair  of  prisms  so  adjusted,  and  perpendicular  to  their  principal  sections,  i.  e.  to  the 
plane  of  the  paper.  Then,  after  refraction  through  both,  it  will  be  seen  by  an  eye  at  E,  as  a  real  object  o'  p', 
having  its  length  o  unaltered,  but  magnified  in  breadth.  Now,  if  we  add  a  second  pair  of  prisms,  similar  to  the 
first,  and  similarly  disposed  with  respect  to  each  other,  so  as  to  form  a  second  achromatic  combination,  but 
having  the  plane  of  their  principal  sections  at  right  angles  to  the  former,  producing  a  refraction  perpendicular  to 
the  plane  of  the  paper,  or  parallel  to  the  length  of  the  distorted  square,  this  will  be  in  like  manner  seen  as  a  real 
and  colourless  object,  but  again  distorted,  its  side  o'  p'  remaining  unaltered,  but  o'  being  magnified.  Thus,  by 
the  effect  of  the  first  distortion,  the  breadth  of  the  square  is  magnified,  and,  by  that  of  the  second,  its  length, 
and  in  the  same  ratio  ;  and  therefore  the  final  result  will  be  an  image  undistorted,  achromatic,  and  magnified. 
The  writer  of  these  pages  had  the  pleasure  of  witnessing  the  very  good  performance  of  one  of  these  singular 
telescopes,  magnifying  about  four  times  in  the  hands  of  its  inventor,  at  Modena,  in  1826.  It  is  evident,  that,  by 
superposing  several  such  telescopes  on  each  other,  the  magnifying  power  may  be  increased  in  geometrical  pro- 
gression. It  is  equally  clear,  that,  by  using  prisms  of  two  different  media  to  form  the  several  binary  combina- 
tions, the  subordinate  spectra  may  be  made  to  counteract  the  secondary  spectra,  arising  from  the  difference  in 
the  scales  of  dispersion  in  the  two  media  ;  and  thus  an  achromaticity,  almost  mathematically  perfect,  might  be 
obtained.  It  is  worthy  of  consideration,  whether,  for  the  purpose  of  viewing  very  bright  objects,  as  the  sun, 
for  instance,  this  species  of  telescope  might  not  prove  of  considerable  service.  It  would  have  the  advantage  of 
being  its  own  darkening  glass,  of  not  bringing  the  rays  to  a  focus,  and  therefore  of  requiring  no  extraordinary 
care  in  the  figuring  of  the  surfaces  ;  and,  in  short,  of  being  exempt  from  all  those  inconveniencies  which  oppose 
the  perfection  of  telescopes  of  the  usual  constructions,  as  applied  to  this  particular  object. 

i  :\A  Proposition.  To  find  the  conditions  of  achromaticity  when  several  prisms  of  different  media  refract  a  ray  of 

Conditions    white  light,  supposing  all  their  refracting  angles  very  small,  and  the  ray  to  pass  nearly  at  right  angles  to  the 
of  achroma-  principal  section  of  each. 

ticity  for  The  refracting  angles  being  A,  A',  A",  &c.,  and  the  refractive  indices  ft,  ft1,  &c.,  the  several  partial  deviations 

several         wjj[  be  D  =  (/»  —  1)  A  ;   D'  =  (jJ  —  1)  A',  &c.;  and  their  sum,  or  the  total  deviation,  will  be  (/i  —  1)  A  + 
small  'angles  0*'  —  1)  A'  +  (/'  —  1)  A"  +  &c.     In   order  that  the  emergent  ray  may  be  colourless,  this  must  be  the  same 
for  rays  of  all  colours  ;  and  its  variation,  when  ft,  ft',  &c.  are  made  to  vary,  must  vanish,  or 


+  &c.  =0. 
Now,  by  equation  (d)  of  Art.  439,  we  have  3  p,  (or,  in  the  notation  of  that  article,  /»  —  p0) 


Therefore  the  above  equation  gives,  when  arranged  according  to  powers  of  %x, 

0  =  |A  C"0  ~  O  «  +  A'  0/0  -  1)  a'  +  A"  (/'„  -  1)  a"  +  &c. 

+  JA  (ft,  -  1)  6  +  A'  (f!0  -  1)  V  -\   A"  G»»0  -  1)  6"  +  &c-  j 

+   &c. 

taking  a',  6',  &c.  to  represent  the  dispersive  powers  of  the  various  orders  for  tha  second  prism,  *",  b",  &c.  for 
the  third,  and  so  on.  Hence,  in  order  that  this  may  vanish  for  all  the  rays  in  the  spectrum,  we  must  have 
(putting,  for  brevity,  /*  for  ftg,  ft'  for  /»'„,  &c.) 


LIGHT.  421 

light.  (^  -  1)  .  A  a  +  (jJ  -  1)  A' a'  +  (/'  -  1)  A"  a"  +  &c.  =  0  -|  _  P««  II. 

— Y— '  0.  -  1)  .  A  6  +  (/»'  -  1)  A'  V  +  (/'  -  1)  A"  b"  +  &c.  =  0  I 

0*  -  1)  .  Ac  +  0*'  -  1)  A'  <?  +  (/'  —  1)  A"  c"  +  &c.  =  0  > 
&c.  &c.  &c.  &c. 

and  so  on.  Generally  speaking,  the  number  of  these  equations  being  infinite,  no  finite  number  of  prisms  can 
satisfy  them  all ;  but  if  we  attempt  only  to  unite  as  many  rays  in  the  spectrum  as  there  are  prisms,  which  is  the 
greatest  approach  to  achromaticity  we  can  attain,  we  have  as  many  equations  as  unknown  quantities,  minus  one, 
and  the  ratios  of  the  angles  to  each  other  become  known.  Thus,  to  unite  two  rays  two  media  suffice,  and  we 
can  only  take  into  consideration  the  first  order  of  dispersions,  which  give 

Cp_l)Aa+</-l)AV*0;~  =  -j£_I  .  -£-.  (J) 

To  unite  three  rays  we  have 

0»  -  1)  A  a  +  (X  -  1)  A'  «'  +  (/."  -  1)  A"  a"  =  0 

Ot  -  1)  A  6  +  (ft  -  1)  A'  b'  +  (//'  -  1)  A"  b"  -  0 
whence  by  elimination 

A'  in  -  1          a  b"  —  b  a '       A"  /.  -  1          a  b'  —  b  a' 


A    '  /-I     '    a'b"-b'a"      A  /."-I"    a1'  b'  -  b"  a 


(*) 


and  so  on  for  any  number. 

In  the  case  of  two  media,  if  the   quantities  6,  c,  &c.  be  not  known,  the  dispersive  powers  of  the  first  order,      453. 
a,  a',  should  be  determined,  not  by  comparison  of  the  extreme  red  and  violet  rays,  which  are  too  little  luminous  Cane  of  two 
to  render  their  strict  union  a  matter  of  importance  ;    we  should  rather  endeavour  to  unite  those  rays  which  are  media- 
at  once  powerfully  illuminating,  and  differing  much  in  colour,  such   as  the  rays  D  and  F.     The  exact  union  of  Best  rays 
these  will  insure  the  approximate  union  of  all  the  rest  better,  on  the  whole,  than  if  we  aimed  at  uniting  the  to  unite. 
extremes  of  the  spectrum,  and  a  far  greater  concentration  of  light  will  be  produced.     This  should  be  carefully 
borne  in  mind  in  all  experiments  on  the  dispersions  of  glass  to  be  used  in  the  construction  of  telescopes. 

If  we  would  produce  the   greatest  possible  achromaticity  by  three  prisms,  the  rays  to  be  selected  for  deter-      454. 
mining  the  values  of  a,  6,  a',  &',  should  be  C,  E,  and  G ;  or,  which  would,  perhaps,  be  still  better,  C,  P,  and  a  Best  rays  to 
ray  half  way  between  D  and  E  ;  but  the  want  of  a  sufficiently  well  marked  line  in  that  part  of  the  spectrum  umte  '" 
throws  some  slight  difficulty  in  the  way  of  this  latter  combination,  when  solar  light  is  used,  and  would  oblige  us  ^jj°  ' 
to  have  recourse  to  some  other  method  of  measurement,  of  which  a  variety  might  be  suggested. 

In  the  case  of  three  media,  if  the  numerators  and  denominators  of  the  expressions  (A)  vanish,  or  nearly  so,  the      455. 
solutions  become  illusory,  or  at  least  inapplicable  in  practice.     This  happens  whenever  either  of  the  fractions  Cases  in 

which  the 
a        a         a'  b  b  b'  formula; 

~T«  ~~JT'  — rr  becomes  equal  to  either  of  the  corresponding  fractions  —-:-,  —rr,  or  —— .      Hence,   to  obtain  become  in- 

a       a          a  b'        b"  b  applicable 

practicable  combinations,  it  is  necessary  to  employ  media  which  differ  as  much  as  possible  in  their  scales  of  dis-  to  pra' 
persive  powers,  i.  e.  in  which  the  coloured  spaces  differ  as  far  as  possible  from  proportionality  ;    such,  for 
instance,  as  flint  glass,  crown  glass,  and  muriatic  acid ;  or,  still  better,  oil  of  cassia,  crown  glass,   and   sulphuric 
acid,  &c. 

§  II.  Of  the  Achromatic  Telescope. 

In  the  refracting  telescopes  described  in  Art.  380,  &c.  the  different  refrangibility  of  the  differently  coloured  rays      456. 
presents  an  obstacle  to  the  extension  of  their  power  beyond  very  moderate  limits.     The  focus  of  a  lens  being  Chromatic 
shorter  as  the  refractive  index  is  greater,  it  follows,  that  one  and  the  same  lens  refracts  violet  rays  to  a  focus  ab<yatlor> 
nearer  to  its  surface  than  red.     This  is  easily  seen  by  exposing  a  lens  to  the  sun's  rays,  and  receiving  the  con-  exp  " 
verging  cone  of  rays  on  a  paper  placed  successively  at  different  distances  behind  it.     At  any  distance  nearer  to 
the  lens  than  its  focus  for  mean  rays,  the  circle  on  the  paper  will  have  a  red  border,  but  beyond  it  a  blue  one ; 
for  the  cone  of  red  rays  whose  base  is  the  lens,  envelopes  that  of  violet  within  the  focus,  its  vertex  lying  beyond 
the  other,  but  is  enveloped  by  it  without,  for  the  converse  reason.     Hence,  if  the  paper  be  held  in  the  focus  for 
mean  rays,  or  between  the  vertices  of  the  red  and  violet  cones,  these  will  then  form  a  distinct  image,  being  col- 
lected in  a  point :  but  the  extreme,  and  all  the  other  intermediate  rays,  will  be  diffused  over  circles  of  a  sensible 
magnitude,  and  form  coloured  borders,  rendering  the  image  indistinct  and  hazy.     This  deviation  of  the  several 
coloured  rays  from  one  focus  is  called  the  "  chromatic  aberration." 

The  diameter  of  the  least  circle  within  which  all  the  coloured  rays  are  concentrated  by  a  lens  supposed  free      457. 
from  spherical  aberration  is  easily  found.     Thus,  in  fig.  107,  if  v  be  the  focus  for  violet,  and  r  for  red  rays,  n  m  o  Least  circle 

of  chromatic 

will  be  the  diameter  of  this  circle.     Now,  by  similar  triangles,  n  o  =  A  B  . ,  and  also  n  o  =z  A  B          —  •  aberration- 

Cc  C  r  '  F'K- 1(>" 


422 


LIGHT 


Ijght. 
j-    -,_-  therefore  equating  these 


sequently  m  r  =  r  v  . 


m  v 


Cr 


m  r 

~c7 


Cv 


«_/  v 

-,  and  m  v  =  m  r  .  — — ,  mv  +  mr  =  mr  . 
C  r 


Cr  +  CD 
~~Cr 


con- 


Cr 


Cr+  Co 


2  Cr  —  rv 


comparison  with  the  whole  refraction.     Therefore  n  o  = 


AB 


very  nearly,  since   the  dispersion  is  small  in 


Now,    f  being    the   reciprocal    focal 


Cr 


1  2  f 

distance  (=  L  +  D  =  (u  —  1)  (R1  —  R")  +  D)  we  have  r  v  =  —  S  —?•  =  -^~  =  - 

fs  T  1 

=   .  — —  and  C  r  =    —y,  supposing  /t  to  represent  the  index  of  refraction  for  extreme  red  rays. 

L          >•• 

Hence  we  get  diameter  of  least  circle  of  chromatic  aberration  =  semi-aperture  X  —f  . 


f    • 


—  semi-aperture  X  dispersive  index  X   ~-r-  ; 


458. 

Use  of  very 
long  tele- 
scopes. 


459. 

Principle  of 
•Jie  achro- 
matic 
telescope. 


General 
equations 
of  achroma- 
ticity. 


and  for  parallel  rays,  when  L  =  f,  simply  semi-aperture  X  dispersive  index. 

Carol.  Hence  the  circle  of  least  colour  has  the  same  absolute  linear  magnitude  whatever  be  the  focal  length  of 
the  lens,  provided  the  aperture  be  the  same.  Now,  in  the  telescope,  the  magnifying  power,  or  the  absolute  linear 
magnitude  of  the  image  viewed  by  a  given  eye-glass,  increases  in  the  ratio  of  the  focal  length  of  the  object-glass, 
(382.)  Therefore,  by  increasing  the  focal  length  of  an  object-glass  without  increasing  its  aperture,  the  breadth 
of  the  coloured  border  round  the  image  of  any  object  diminishes  in  proportion  to  the  image  itself,  and  thus  the 
confusion  of  vision  is  diminished,  and  the  telescope  will  possess  a  proportionally  higher  magnifying  power.  In 
consequence  of  this  property,  before  the  invention  of  the  achromatic  telescope,  astronomers  were  in  the  habit  of 
using  refracting  telescopes  of  enormous  length,  even  so  far  as  100  or  150  feet;  and  Huyghens,  in  particular, 
distinguished  himself  by  the  magnitude  and  excellent  workmanship  of  his  glasses,  and  by  the  important  astrono- 
mical discoveries  made  with  them. 

The  achromatic  object-glass,  however,  by  enabling  us  to  reduce  the  length  of  the  telescope  within  more  reason- 
able bounds,  has  rendered  it  a  vastly  more  manageable  and  useful  instrument.  To  conceive  its  principle,  we  have 
only  to  recur  to  what  has  already  been  said  in  Art.  451 — 454,  respecting  achromatic  prisms.  A  lens  is  nothing 
more  than  a  system  of  infinitely  small  prisms  arranged  in  circular  zones  round  a  centre,  with  refracting  angles 
increasing  as  their  distance  from  the  centre  increases,  so  as  to  refract  all  the  rays  to  one  point ;  and  if  we  can 
achromatise  each  elementary  prism,  the  whole  system  is  achromatic.  The  equations  (J)  apply  at  once  to  this 
view  of  the  structure  of  a  lens.  For,  suppose  R',  R"  to  be  the  curvatures  of  the  two  surfaces  of  the  first  lens, 
L'  its  power,  and  ft'  its  refractive  index,  then,  for  a  given  aperture,  or  at  a  given  distance  from  the  centre, 
R'  —  R/r,  the  difference  of  the  curvatures,  expresses  the  angle  made  by  tangents  to  the  surfaces,  or  the  refracting 
angle  of  the  elementary  prism  ;  or  R'  —  R"  =  A1;  and  similarly  for  the  other  lenses,  A''  =  R"1  —  R'T,  and  so 
on,  so  that  the  equations  become 

(X-  1)  (RJ  -  R")  .  a'  +  (X-l)  (R'"--  Rlv)  a"  +  &c.  =  0  &c. ; 
that  is  simply 

L' .  a'  +  L"  .  a"  +  L'" .  a'"  +  &c.  =  0 


L' .  6'  +  L"  .  I"  +  L'"  .  b'"  +  &c.  =  0 

L' .  </  +  L"  .  c"  +  L'"  .  d"  +  &c.  =  0 

&c. 


(a) 


460.  These  equations  afford  all  the  relations  necessary  to  insure  achromaticity ;  and  when  satisfied,  since  they  do 
Otherwise  not  contain  D,  they  show  that  an  object-glass  "which  is  achromatic  for  any  one  distance  of  the  object  is  so  for 
deduced.  an  distances.  It  is  evident,  that  the  same  system  of  equations  may  be  obtained  directly  from  the  expression  in 

Art.  265  for  the  joint  power  of  a  system  of  lenses  whose  individual  powers  are  L',  L",  &c.     For  the  condition 

of  achromaticity  gives  1  L  —  0,  that  is 

5L'  +  5L"  +  SL"'  +&c.  =  0. 


But  since  L'  =  (X  -  1)  (R'  —  R")  &c.  (according  to  the  system  of  notation  there  adopted) 

«L'=  (R'-  R")  SX  = 


V 

/  - 


But  in  the  equation  (d)  if  we  put  in  succession  for  ft0  the  values  //,  //',  &c.,  for  /» —  /*0  respectively,  */»'»  *  p",  &c., 
and  for  a,  b,  &c.  the  systems  of  coefficients  a',  b',  &c. ;  a",  b",  &c. ;  and  suppose _  °     =  p,  we  shall  have 


Part  II. 


=  a"  p  +  b"p*+  &c; 


LIGHT.  423 

and  therefore  v         _/ 

0  =  I/  {  a!  p  +  b'p*  +  &c.  }  .+  L"  {  al'p  +  V  p*  +  &c.  }  +  &c.  — v— 

which,  being  made  to  vanish  independently  of  p,  gives  the  very  same  system  of  equations  as  (a.) 

To  satisfy  all  these  equations  at  once  with  any  finite  number  of  lenses  being  impossible,  we  must  rest  content 
with  satisfying  as  many  of  the  most  important  as  the  number  of  lenses  will  permit.     Thus,  if  we  have  two  lenses  Object  glass 
of  different  media,  such  as  flint  and  crown  glass,  for  instance,  one  only  of  them  can  be  satisfied,  and  this  must  ^ed"° 
of  course  be  the  first,  viz. 

Ua'  +  L»a»=0,         BrH.es-    'j  (6) 

which  shows  that  the  powers  of  the  lenses  must  oppose  each  other,  and  be  to  each  other  inversely  (and  of  course 
their  focal  lengths  directly)  as  the  dispersive  powers.  In  such  a  combination,  the  values  of  a',  a",  the  dispersive 
powers,  however,  ought  not  to  be  obtained  from  the  relative  refractions  for  the  extreme  red  and  violet  rays  of  the 
spectrum,  (according  to  the  remark  in  Art.  453,)  but  rather  from  the  strongest  and  brightest  rays  whose  colours 
are  in  decided  contrast;  such,  for  instance,  as  the  rays  C  and  F  in  Fraunhofer's  scale. 

With  three  lenses  of  different  media,  two  of  the  equations  of  achromaticity  can  be  satisfied,  and  the  secondary      462. 
spectrum  corrected,  thus  we  have  of  {J^e  a 

a'  b'"  -  b'  a"'   -)  media. 


0  =  !/«'  +  L"  a"  +  L'"  .   ,         -   _ 

0  =  L'  6'  +  L"  6"  +  L'"  b'" 


and  in  determining  the  values  of  a',  b',  &c.  the  rays  to  be  employed  should  be  the  brightest  yellow  for  a  middle 
ray,  and  a  pretty  strong  red  and  blue  for  the  extremes.  The  rays  B,  E,  H  are  perhaps  inferior  to  C,  E,  G  for 
this  purpose. 

Hence  in  a  double  object-glass  having  a  positive  focus  the  least  dispersive  lens  must  be  of  a  convex  or  positive, 
and  the  most  so  of  a  negative,  or  concave  character.  The  order  in  which  they  are  placed  is  of  no  consequence, 
as  far  as  achromaticity  is  concerned. 

A  single  lens,  as  we  have  seen,  neither  admits  of  the  destruction  of  the  spherical,  nor  chromatic  aberration, 
(Art.  296  and  457  ;)  but  if  we  combine  two  or  more  lenses  of  different  media,  the  equations  s,  t,  u,  v  of  Art.  Sl 
309,  310,  312,  and  313,  combined  with  the  equations  just  derived  (a).  Art.  459,  or  so  many  of  them  as  are  not  "ter°"(io|j~  Oi 
incompatible,  afford  us  the  means  of  annihilating  both  species  of  aberration  at  once ;  and  what  is  curious,  and  both  aberra- 
must  be  regarded  as  singularly  fortunate,  the  relations  afforded  by  the  destruction  of  the  chromatic  aberration,  tions. 
which,  at  first  sight,  would    appear  likely  greatly  to  complicate  the  inquiry,  tend,  on  the  contrary,  remarkably 
to  simplify  it,  being  in  fact  the  very  relations  the  analyst  would  fix  upon  to  limit  his  symbols,  and  give  his  final 
equations  the  greatest  simplicity  their  nature  admits,  if  left  at  his  disposal.     For,  it  will  be  remarked,  that  in  the 
general  equation  for  the  destruction  of  the  spherical  aberration,  A  /  =  0,  or 

0  =  ~  («'  -  p  D'  +  y  D")  +  ¥-  (a'"  +  ft"  D"  +  7"  D"«)  +  &c. ;  (d) 

the  expressions  within  the  parentheses  are  all  of  the  second  degree  when  expressed  in  terms  of  the  curvatures  of 
the  surfaces,  and  of  D'  =  D  the  proximity  of  the  radiant  point  to  the  first  lens ;  and  as  L',  L",  &c.  are  respec- 
tively of  the  first  degree,  in  terms  of  the  curvatures,  the  whole  is,  in  its  general  form,  of  the  third  degree,  and 
the  equation  of  a  cubic  form.  But  the  conditions  of  achromaticity,  which  assign  relations  only  between  L',  L", 
&c.  without  involving  R',  II",  &c.  enable  us  to  eliminate  these  quantities  and  replace  them  in  the  above  equation, 
by  giving  combinations  of  a',  a",  b',  b",  &c.,  so  that  it  becomes  reduced  to  a  quadratic  form,  and  its  treatment 
simplified  accordingly. 

Let  us  proceed  now  to  develope  the  equation  (d),  in  which,  according  to  the  foregoing  remark,  when  the  con-       465. 
ditions  of  achromaticity  are  introduced,  I/,  L",  &c.  may  be  regarded  as  given  quantities ;  for,  taking  L  =  L'  +  Determina- 
L"  +  &c.  =  the  power  of  the  compound  lens,  (which  we  may  suppose  given,  or,  if  we  please,  assume  equal  to  tl( 
unity,)  this,  combined  with  the  equations  (a),  determines  the  values  of  L',  &c.     Thus,  in  the  case  of  two  lenses,  {^"several 

a1  L  ®  L  lenses. 

if  we  put  TT  for  the  ratio  of  the  dispersive  powers,  or  CT  =  —^-  we  have  L'  =  — — — • ,  L"  =  —  ^  _  CT ;    and 

similarly  for  three  or  more  lenses.     Suppose  then  we  represent  by  r1,  T",  /",  &c.  the  respective  curvatures  of  the 

first,  or  anterior  surfaces  of  the  first,  second,  third,  &c.  lens,  in  order ;  the  first  being  that  on  which  the  rays  first 

j ,  Develope- 

fall.     Then  we  have  L'  =  (/  -  1)  (R'  —  R")  =  (ji1  -  1)  (r7  -  R",)  so  that  R"=  r1 j— y- ;  and  similarly  m^^  the 

•^11  equation. 

Rlv  =  r"  —  — ft — T~»  &c-     We  must  therefore  put  in  the  foregoing  expressions 

R^/;         R"  =  /  - —^  ;         R"'=r";         R"  =  r"  -     ,,U      ,  Ac. 


424  LIGHT. 

Ughx      Hence  by  substitution  of  these  in  the  values  of  a,  /3,  &c.  (Art.  293)  we  get 

i,' 
a'  =  (2  +  p 


.(8X+1).7=_.I/, 


VX-i 


L. 


p  =  (4  +  4  X)  i*  —  (3  X  +  O  • 

y  =  2  +  3  /, 


X-l 


u 


and  similarly  for  a",  /3",  7",  &c.  So  that,  substituting  again  these  expressions,  and  putting  for  D"  its  equal 
L'  +  D',  for  D'"  its  equal  L'  +  L"  +  D',  and  so  on,  we  have,  finally,  for  the  general  equation  A  /  =  0,  as 
follows  : 


o=  {(y  +  I) 


LV 


4r) 


D. 


L  +  3)L,,,  +  (A  + 

_  4  /('i  +  -L)  L'r   +  (l  -h-4-^)  L'V  +  (l  +    V)  L'"r'"  +  &c-  [ 

I  V  /»'   /  V  X    '  >  r*    ' 


+  1 


,.-1       L'"2+&C-} 


+  2  J(-i-    +  3)  L'L"  +  (-^r    +  3)  (L'  +  I/O  L'"  +  &c.j 
+  D  •  j  (-4-   +  3)  L'  +  (-^-  +  s)  L"  +  (-4r  +  3)  L'"  +  &c.  J. 


466.  For  brevity,  let  us  represent  by  X,  the  terms  of  this  expression,  independent  of  the  quantity  D  ;    by  Y,  the 

assemblage  of  terms  multiplied  by  D';  and  by  Z,  those  multiplied  by  D'3,  and  we  shall  have 


A/=  - 


Y.D+Z.D'}; 


467. 

The  distinc- 
tion of  aber- 
ration an  in- 
determinate 
problem. 

Conditions 
limiting  it. 
Cliiraut's. 


and  if  this  vanish  the  aberration  is  destroyed.  Now,  first,  if  we  regard  only  parallel  rays,  or  suppose  D  =  o. 
this  reduces  itself  to  X  =  0,  so  that  the  condition  X  =  0  being  satisfied,  the  telescope  will  be  perfect  when  used 
for  astronomical  purposes,  or  for  viewing  objects  so  distant  that  D'  may  be  disregarded. 

The  equation  X  =  0  is  of  the  second  degree  in  each  of  the  quantities  r1  r",  &c.,  whose  number  is  that  of  the 
lenses.  Consequently,  this  condition  alone  is  not  sufficient  to  fix  their  values  ;  and,  without  assuming  some 
further  relations  between  them,  or  some  other  limitations,  the  problem  is  indeterminate,  and  the  aberration  may 
be  destroyed  in  an  infinite  variety  of  ways.  Confining  ourselves  at  present  to  the  consideration  of  two  lenses 
only,  since  X  =  0  contains  only  two  unknown  quantities,  one  other  equation  only  is  required,  and  we  have  only 
to  consider  what  other  condition  will  be  attended  with  the  greatest  practical  advantages.  Clairaut  has  proposed 
to  adjust  the  two  lenses  so  as  to  have  their  adjacent  surfaces  in  contact  throughout  their  whole  extent,  to  allow 
of  their  being  cemented  together,  and  thus  avoid  the  loss  of  light  by  reflection  at  these  surfaces.  This  certainly 
would  be  a  great  advantage  were  it  possible  so  to  cement  two  glasses  of  large  size  together,  as  to  bring  neither 
of  them  into  a  state  of  strain  as  the  cement  cools,  or  otherwise  fixes  ;  and  were  it  not  for  the  further  incon- 
venience, that  the  media  being  of  course  differently  expansible  by  heat,  every  subsequent  change  of  temperature 
would  necessarily  distort  their  figure,  as  well  as  strain  their  parts,  when  thus  forcibly  held  together,  just  as  we 
see  a  compound  lamina  of  two  differently  expansible  metals  assume  a  greater  or  less  curvature,  according  to 
the  temperature  it  is  exposed  to.  Meanwhile  the  condition  in  question  is  algebraically  expressed  by  I/=  (p1  —  1) 
(r'  —  i")  ;  for  in  this  case  R'  =  r',  and  R"  =  R"'  =  r",  and  this  being  of  the  first  degree  only  in  r1,  r'1,  affords  a 
final  equation  of  a  quadratic  form  by  elimination  with  X  =  0,  which  latter,  in  the  case  before  us  of  two  lenses, 
is  the  same  as  the  equation  (c),  Art.  312,  writing  only  r'  for  R',  and  r"  for  R'". 


Part  II. 


LIGHT.  4'25 

Light.          But  this  condition  of  Clairaut's  has  another  and  much  greater  inconvenience,  which   is,  that   the  resulting     Part  II. 
— N^— '  quadratic  has  its  roots  imaginary,  when  the  refractive  and  dispersive  powers  of  the  glasses  are  such  as  are  by  no  v~"Tfi<T"" 
means  unlikely  to  occur  in  practice;  and  without  the  limits  of  refraction  and  dispersion,  for  which  they  are  real,      £ b&- 
the  resulting  curvatures  change  so  rapidly  on  slight  variations  of  the  data,  as  to  make  their  computation  delicate,  bert.gem 
and  interpolation  between  them,  so  as  to  form  a  table,  very  troublesome.     D'Alembert,  in  his  Opuscules,  torn,  iii., 
has  proposed  a  variety  of  other  limitations,  such,  for  instance,  as  annihilating  the  spherical  aberration  for  rays 

*  v  X  Y 

of  all  colours,  (which  comes  to  the  same  as  supposing  at  once  X  ==  0  and  — — p  J//  -\ jf-  '  /»"    =   0,   and 

which  leads  to  biquadratic  equations,  and  affords  no  practical  advantage,)  &c.     But,  without  going  into  useless 
refinements  of  this  kind,  the  very  form  of  the  general  equation  X  +  Y.D'  +  Z  .  D'2  =  0  points  out  a  condition 
combining  every  advantage  the  case  is  susceptible  of.     This  consists  in  putting  Y  =  0.     By  this  supposition,  the 
term  depending  on  D'  is  destroyed,  without  assuming  D'=  0  ;  so  that  the  telescope  is  not  only  perfect  for  parallel  Another 
rays,  but  admits  of  as  considerable  a  proximity  of  the  object  without  losing  its  aplanatic  character,   as   the  proposed. 
nature  of  the  case  will  allow.     The  term  Z  .  D'2  indeed,  or 


cannot  vanish  when  two  lenses  only  are  used,  being  composed  wholly  of  given  functions  of  the  refractive  and 
dispersive  powers,  unless  by  D'  itself  vanishing,  or  by  an  accidental  adjustment  of  the  values  of  fJ,  p",  L',  &c. 
But  except  the  object  be  brought  within  a  comparatively  small  distance  from  the  telescope,  (such  as  ten  times  its 
own  length,)  the  square  of  D'  is  always  so  small  as  to  allow  of  our  disregarding  this  term,  and  considering  the 
instrument  as  perfectly  aplanatic  when  Y  =  0.  Now  this  equation,  being  of  the  first  degree  in  r1,  i",  adds  no 
new  algebraic  difficulty  to  the  problem,  but  leads  by  elimination  to  a  final  quadratic  ;  and,  what  is  of  most  con- 
sequence, for  such  values  of  pi,  /u.",  and  the  dispersive  ratio  w  as  occur  in  practice,  the  roots  of  this  quadratic 
are  always  real,  and  the  resulting  curvatures  of  all  the  surfaces  are  moderate,  and  well  adapted  for  practice  ;  more 
so,  indeed,  than  in  any  construction  hitherto  proposed.  They  are,  moreover,  such  as  to  afford  remarkable  and 
peculiar  facilities  for  interpolation,  as  we  shall  presently  see.  These  reasons  seem  to  leave  no  room  for  hesita- 
tion in  fixing  on  the  condition  Y  =  0,  as  that  which  ought  to  be  introduced  to  limit  the  problem  of  the  con- 
struction of  a  double  object-glass,  and  to  render  it,  so  far  as  it  can  be  rendered,  aplanatic. 

This  equation,  in  the  case  in  question,  is  469 


0  = 

3X  +  1 
/-> 

which  is  to  be  combined  with  (v).  Art.  412,  in  which  R'  =  /  and  R'"=  r".  To  reduce  these  to  numbers,  /»',  ft"  470 
and  the  dispersive  ratio  CT  must  first  be  known.  The  readiest  and  most  certain  way  in  practice,  for  the  use  of 
the  optician,  is  to  form  small  object-glasses  from  specimens  of  the  glasses  intended  to  be  employed,  and  by  trial 
work  them  till  the  combination  is  as  free  from  colour  as  possible,  by  the  test  usually  had  recourse  to  in  practice. 
This  is,  to  examine  with  a  high  magnifying  power  the  image  of  a  well  defined  white  circle,  or  circular  annulus  on 
a  black  ground.  If  its  edges  are  totally  free  from  colour,  the  adjustment  is  perfect,  but  (owing  to  the  secon- 
dary spectrum)  this  will  seldom  be  the  case  ;  and  there  will  generally  be  seen  on  the  interior  edge  of  the  annulus 
a  faint  green,  and  on  the  exterior  a  purplish  border,  when  the  telescope  is  thrown  a  little  out  of  focus  by  bringing 
the  eye-glass  too  near  the  object-glass,  and  vice  vend.  The  reason  is,  that  while  the  gr<;at  mass  of  orange  and 
blue  rays  is  collected  in  one  focus,  the  red  and  violet  are  converged  to  a  focus  farther  from,  and  the  green  to 
one  nearer  to  the  object-glass  ^  the  refraction  of  the  green  rays  being  in  favour  of  the  convex  or  crown  glass,  and 
of  the  red  and  violet  (which  united  form  purple)  in  favour  of  the  flint  (see  table,  Art.  443)  or  concave  lens.  The 
focal  lengths  of  the  lenses  are  then  to  be  accurately  determined,  and  the  ratio  of  the  dispersions  (or)  will  then 
be  known,  being  the  same  with  that  of  the  focal  lengths  (454).  The  refractive  indices  will  be  best  ascertained 
by  direct  observation,  forming  portions  of  each  medium  into  small  prisms.  Now,  CT  being  known,  if  we  take 

unity  for  the  power  of  the  compound  lens,  we  have  I/  z=  —      —  and  L"  =  —          — ,   so   that  L/  and  L"  are 

known,  and  we  have  therefore  only  to  substitute  their  values  and  those  of  //,  ft",  in  the  algebraic  expressions, 
and  proceed  to  eliminate  by  the  usual   rules.     The  following   compendious  table  contains  the  result  of  such  Dimensions 
calculations  for  the  values  of  ft1,  ft."  and  to-  therein  stated,  together  with  the  amount  of  variation  produced   by  °f an  ap'a- 
varying  either  of  the  refractive  indices  independently  of  the  other,  for  the  sake  of  interpolation  by  proportional  nat'CODJcct- 
parts.     Fig.  108  is  a  representation  of  the  resulting  object-glass.  glass' 


426 


LIGHT. 


Ligbt 


Table  for  finding  the  Dimensions  of  an  Aplanatic  Object-glass. 

Refractive  index  of  crown,  or  convex  lens  =/»'=:  1.524. 

Refractive  index  of  flint,  or  concave  lens  =:  //'  =  1.585. 

Compound  focal  length  =  10.000. 


PartFI. 


CROWN  LENS. 

FLINT  LENS. 

Second 

Third 

First  surface,  convex. 

surface, 

Surface, 

Fourth  surface,  convex. 

convex. 

Concave. 

Variation  of 

Variation  of 

Variation  of 

Variation  of 

Dis- 

radius for  a 

radius  for  a 

Focal 

Radius  for 

radius  for  a 

radius  for  a 

per- 
sive 

above  re- 

change  of 
+  0.010  in 

change  of 
+  0.010  in 

Radius 
of  con- 

length 
of 

Radius 
of  con- 

the above 
refractive 

change  of 
+  0.010  in 

change  of 
+  0.010  in 

Focal 
length  of 

ratio 

ref.  index  of 

ref.  index  of 

vexity. 

crown 

cavity. 

indices. 

ref.  index  of  ref.  index  of 

flint  lens. 

tr  =. 

crown  glass. 

flint  glass. 

lens. 

crown  glass. 

flint  glass. 

0.50 

6.7485 

+  0.0500 

-  0.0030 

4.2827 

5.0 

4.1575 

14.3697 

+  0.9921 

-  0.3962 

10.0000 

0.55 

6.7184 

+  0.0740 

-  0.0011 

3.6332 

4.5 

3.6006 

14.5353 

+  1.0080 

-0.5033    8.1818 

0.60 

6.7069 

+  0.0676 

+  0.0037 

3.0488 

4.0 

3.0640 

14.2937 

+  1.1049 

—  0.5659 

6.6667 

0.65 

6.7316 

+  0.0563 

+  0.0125  2.5208 

3.5 

2.5566 

13.5709 

+  1.1614-0.6323 

5.3846 

0.70 

6.8279 

+  0.0335 

+  0.0312  2.0422 

3.0 

2.0831 

12.3154 

+  1.1613  —0.7570    4.2858 

0.75 

7.0816 

—  0.0174 

+  0.0568 

1.6073 

2.5 

1.6450 

10.5186 

+  1.0847  -0.7207    3.3333 

the  table. 


To  apply  this  table  to  any  other  proposed  state  of  the  data,  we  have  only  to  consider  that  to  compute  the  radius 
of  any  one  of  the  surfaces,  as  the  first  or  fourth,  we  have  only  to  regard  each  element  as  varying  separately,  and 
471.       take  proportional  parts  for  each.     The  following  example  will   elucidate  the  process  :    Required  the  dimensions 
Example  of  for  an  object-glass  of  30  inches  focus,  the  refractive  index  of  the  crown  glass  being  1.519,  and  that  of  the  flint 
»  »"w»°f    1-589;    the  dispersive  powers  being  as  0.567  :  1,   or    0.567  being  the   dispersive   ratio.      Here  p.'  =  1.519, 
ft"  —  1.589,  and  ro  =  0.567.    The  computation  must  first  be  instituted  for  a  compound  focus  =  10.000,  as  in  the 
table,  and  we  proceed  thus : 

1st.  Subtract  the  decimal  (0.567)  representing  the  dispersive  ratio  from  1.000,  and   10  times  the  remainder 
(=  10  x  0.433  =  4.330)  is  the  focal  length  of  the  crown  lens. 

2nd.  Divide  unity  by  the  decimal  above  mentioned,  (0.567,)  subtract  1.000  from  the  quotient  ( = 

v  0.567 

1.7635,  minus  1  =  0.7635)  and  the  remainder  multiplied  by  10  (or  7.635)  is  the  focal  length  of  the  flint  lens. 
We  must  next  determine  by  the  tables  the  radii  of  the  first  and  fourth  surfaces  for  the  dispersive  ratios  there  set 
down  (0.55  and  0.60)  next  less  and  next  greater  than  the  given  one.  For  this  purpose  we  have 


Refractive  powers  given. . . 
Refractive  powers  in  table 


1.519  and  1.589 
1.524    .       1.585 


Differences       —  0.005      +  0.004 

The  given  refraction  of  the  crown  bf.ing  less,  and  of  the  flint  greater,  than  their  average  values  on  which  the 
table  is  founded.  Looking  out  now  opposite  to  0.55  in  the  first  column  for  the  variations  in  the  two  radii 
corresponding  to  a  change  of  +  0.010  in  the  two  refractions,  we  find  as  follows: 

First  surface.      Fourth  surface. 

For  a  change  =  +  0.010  in  the  crown  +  0.0740       +  1.0080 
For  a  change  =  +  0.010  in  the  flint     -  0.0011       —  0.5033 

But  the  actual  variation  in  the  crown  instead  of  +  0.010  being  —  0.005,  and  of  the  flint  +  0.004,  we  must  take 
the  proportional  parts  of  these,  changing  the  sign  in  the  former  case  ;  thus  we  find  the  variations  in  the  first  and 
last  radii  to  be 


LIGHT.  427 

First  surface.  Fourth  surface 
i  in  the  crown       —  0.0370          —  0.5040 

For  +  0.004  variation  in  the  flint    .  .    —  0.0004          -  0.2013 


light.  First  surface.         Fourth  surface.  Part  IL 

—^~s  For  —  0.005  variation  in  the  crown       —  0.0370          —  0.5040  ,,-v-^. 


Total  variation  from  both  causes  ....    —  0.0374          —  0.7053 
But  the  radii  in  the  table  are    6.7184  14.5353 


Hence  the  radii  interpolated  are  ....          6.6810  13.8300 

If  we  interpolate,  by  a  process  exactly  similar,  the  same  two  radii  for  a  dispersive  ratio  0.60,  we  shall  find, 
respectively, 

First  surface.         Fourth  surface. 
For  a  variation  of  —  0.005  in  the  crown  —  0.0338         —  0.5524 

For  a  variation  of  +  0.004  in  the  flint    +0.0015          —0.2264 


Total  variation   —  0.0323          -  0.7788 

Radii  in  table 6.7069  14.2937 


3871 


Interpolated  radii 6.6746  13.5149 

Having  thus  got  the  radii  corresponding  to  the  actual  refractions  for  the  two  dispersive  ratios  0.55   and  0.60, 
it  only  remains  to  determine  their  values  for  the  intermediate  ratios  0.567  by  proportional  parts ;  thus 
First  radius.         Fourth  radius. 

For 0.600  6.6746  13.5149 

For 0.550  6.6810  13.S300  0.050  :  0.567  -  0.050  =  0.017  ::- 0.0064  :- 0.0022 

0.050  0.017:  :- 0.3151: -0.1071 

Differences    +  0.050         —  0.0064          -  0.3151 

So  that  6.6810  —  0.0022  =  6.6788,  and  13.8300  —  0. 1071  =  13.7229,  are  the  true  radii  corresponding  to  the 

given  data.     Thus  we  have,  for  the  crown  lens,  focal  length  =  4.330  =  -—-,  radius  of  first  surface  =  6.6788 

Li 

=  -=77,  index  of  refraction  =  1.519  ==  p!,  whence  by  the  formula  L'  =  (/*'  —  1)  (R'  —  R")  — =rr  radius  of  the 
H,  R 

other  surface  is  —  3.3868.     Again,  for  the  flint  lens,  the  focal  length  =   —rj, —   =*    —    7.635,    radius   of    the 
posterior'  surface  =r  — r-  =  —  13.7729,  index  of  refraction  /u"  =  1.589,  whence  we  find  =  —  3. 

for  the  radius  of  the  other  surface.     The  four  radii  are  thus  obtained  for  a  focal  length  of  10  inches,  and  multi 
plying  by  3  we  have  for  the  telescope  proposed 

in.  in.  in.  jn. 

radius  offirst  surface  =  +  20.0364;     of  second,  —  10.1604  ;     of  third,  -  10.1613;     of  fourth,  —  41.1687. 

Here,  then,  we  see  that  the  radii  of  the  two  interior  surfaces  of  the  double  lens  (fig.  108)  differ  by  scarcely  473. 
more  than  a  thousandth  part  of  an  inch  ;  so  that,  should  it  be  thought  desirable,  they  may  be  cemented  together. 
This  is  not  merely  a  casual  coincidence,  for  the  particular  state  of  the  data ;  if  we  cast  our  eyes  down  the 
table  we  shall  find  this  approximate  equality  of  the  interior  curvatures  (those  of  the  second  and  third  surfaces) 
maintained  in  a  singular  manner  throughout  the  whole  extent  of  the  variation  of  •&.  Thus  the  construction, 
here  proposed  in  reality  for  glasses  of  the  ordinary  materials,  approaches  considerably  to  that  of  Clairaut  already 
mentioned. 

In  order  to  put  these  results  to  the  test  of  experience,  Mr.  South  procured  an  achromatic  telescope  to  be  4/3. 
executed  on  this  construction  by  Mr.  Tulley,  one  of  the  most  eminent  of  our  British  artists,  which  is  now 
in  the  possession  of  J.  Moore,  Esq.  of  Lincoln.  Its  focal  length  was  45  inches,  and  aperture  3J,  and  its  per- 
formance was  found  to  be  fully  adequate  to  the  expectation  entertained  of  it,  bearing  a  magnifying  power  of 
300  with  perfect  distinctness,  and  separating  easily  a  variety  of  double  stars,  &c.  A  more  minute  account 
of  its  performance  will  be  found  in  the  Journal  of  the  Royal  Institution,  No.  26.  Should  the  splendid  example 
set  by  Fraunhofer  be  followed  up,  and  the  practice  of  the  optician  be  in  future  directed  by  a  rigorous  adherence 
to  theory,  grounded  on  exact  measurements  of  the  refractive  powers  of  his  glasses  on  the  several  coloured  rays, 
it  will  become  necessary  to  develope  the  above  table  more  in  detail. 

When  three  media  are  employed  in  the  construction  of  object-glasses,  it  should  be  our  object  to  obtain  as       474 
great  a  difference  as  possible  in  their  scales  of  action  on  the  differently  coloured  rays.     Dr.  Blair,  to  whom  we  Object- 
are  indebted  for  the  first  extensive  examination  of  the  dispersive  powers  of  media  as  a  physical  character,  and  glasses  of 
who  first  perceived  the  necessity  of  destroying  the  secondary  spectrum,  and  pointed  out  the  means  of  doing  it,  ;llree  media 
is  the  only  one  hitherto  who  has  bestowed  much  pains  on  this  important  part  of  practical    optics ;    which, 
considering  the  extraordinary  success  he  obtained,  and  the  perfection  of  the  telescopes  constructed  on  his  prin- 
ciples, is  to  be  regretted.     We  have  no  idea,  indeed,  for  the  reasons  already  mentioned,  that  very  large  object- 

3  K  2 


428  LIGHT. 

Light.       glasses,  enclosing  fluids,  can  ever  be  rendered  available  ;   but  to  render  glasses  of  moderate  dimensions  more 
Vi^-v— '  perfect,  and  capable  of  bearing  a  higher  degree  of  magnifying  power,  is   hardly  less  important  as  an  object  of 
practical   utility.     His  experiments  are  to  be  found   in    the  Transactions  of  the  Royal  Society  of  Edinburgh, 
1791.     We  can  here  do  little  more  than  present  a  brief  abstract  of  them. 

475.  Dr.  Blair  having  first  discovered  that  the  secondary  fringes  are  of  unequal  breadths,  when  binary  achromatic 

Dr.  Blair's  combinations,  having  equal  total  refractions,  are  formed  of  different  dispersive  media,  was  immediately  led  to 
>n  consider,  that  by  employing  two  such  different  combinations  to  act  in  opposition  to  each  other,  if  the  total 
refractions  were  equal,  the  ray  would  emerge  of  course  undeviated,  and  with  its  primary  spectrum  destroyed  ; 
three  media.  ^ut  a  secondary  spectrum  would  remain,  equal  to  the  difference  of  the  secondary  spectra  in  the  two  combina- 
tions. Therefore,  by  a  reasoning  precisely  similar  to  that  which  led  to  the  correction  of  the  primary  spectrum 
itself,  (Art.  426  and  427,)  if  we  increase  the  total  refraction  of  that  combination  A  which,  cieteris  paribns,  gives 
the  least  secondary  spectrum,  its  secondary  colour  will  be  increased  accordingly,  till  it  becomes  equal  to  that  of 
the  other  B  ;  so  that  the  emergent  beam  will  be  free  from  the  secondary  spectra  altogether,  and  will  be  deviated 
on  the  whole  in  favour  of  the  combination  A.  Reasoning  on  these  grounds,  Dr.  Blair  formed  a  compound,  or 
binary  achromatic  convex  lens  A,  (fig.  109,)  of  two  fluids  a  and  6,  (two  essential  oils,  such  as  naphtha  and  oil 
of  turpentine,  differing  considerably  in  dispersion,)  which,  when  examined  alone,  was  found  to  have  a  greater 
refractive  power  on  the  green  rays  than  on  the  united  red  and  violet.  He  also  formed  a  second  binary  lens  B, 
of  a  concave  character,  and  also  achromatic,  (i.  e.  having  the  primary  spectrum  destroyed,)  consisting  of  the  more 
dispersive  oil  (6)  and  glass,  and  in  which  the  green  rays  are  also  more  refracted  than  the  united  red  and  violet, 
but  in  a  greater  degree  in  proportion  to  the  whole  deviation,  than  in  the  other  combination;  and  in  precisely 
the  same  degree  was  the  focal  length  of  this  lens  increased  or  its  refraction  diminished,  when  compared  with 
that  of  the  combination  A.  When,  therefore,  these  two  lenses  were  placed  together,  as  in  fig.  109,  an  excess  of 
refraction  remained  in  favour  of  the  convex  combination  ;  but  the  secondary  spectra  of  each  being  equal  and 
opposite  (by  reason  of  the  opposite  character  of  the  lenses)  were  totally  destroyed.  In  fact,  he  states,  that  in 
a  compound  lens  so  constructed,  he  could  discover  no  colour  by  the  most  rigid  test ;  and  thence  concluded, 
not  only  the  red,  violet,  and  green  to  be  united,  but  also  all  the  rest  of  the  rays,  no  outstanding  colour  of  blue 
or  yellow  being  discernible.  In  placing  the  lenses  together,  the  intermediate  plane  glasses  may  be  suppressed 
altogether,  as  in  fig.  110. 

476  It  was  in  the   course  of  these  researches  that  Dr.  Blair  was  led  to  the  knowledge  of  the  possibility  of  forming 

Hemarkahle  binary  combinations,  having  secondary  spectra  of  opposite  characters  ;    that  is,  in  which  (the  total  refraction 
property  of   lying  the  same  way)  the  order  of  the  colours  in  the  secondary  spectra  should  be  inverted.     In  other  words,  that 
the  muriatic  wnjie  jn  SOme  combinations  the  green  rays  are  more  refracted  than  the  united  red  and  violet,  in  others  they  are 
less  so.     He  found,  for  instance,  that  while  in  most  of  the  highly  dispersive  media,  including  metallic  solutions, 
the  green  lay  among  the  less  refrangible  rays  of  the  spectrum,  there  yet  exist  media  considerably  dispersive,  in 
which   the  reverse  holds  good.     The   muriatic  acid,  among  others,  is  in  this  predicament.     Hence,  in  binary 
combinations  of  glass  with  this  acid,  the  secondary  spectrum  consists  of  colours  oppositely  disposed  from  that 
formed  by  glass  and  the  oils,  or  by  crown  and  flint  glass.     In  consequence  of  this,  to  form  an  object-glass  of 
two  binary  combinations,  as  described  in  the  last  article,  they  must  both  be  of  convex  characters.     But  this  affords 
Dr.  Blair's     no  particular  advantage.     Dr.  Blair,  however,  considered  the  matter  in  another  and  much  more  important  light, 
discovery  of  as  offering  the  means  of  dispensing  with  a  third  medium  altogether,  and  producing  by  a  single  binary  combina- 
***    ,       tion  a  refraction  absolutely  free  from  secondary  colour.     To  this  end  he  considered,  that   it  appears  to  depend 
same  "scale    entire'y  on  t"e  chemical  nature  of  the  refracting  medium,  what  shall  be  the  order  and  distribution  of  the  colours 
ofdispersion  in  the  spectrum,  as  well  as  what  shall  be  the  total  refraction  and  dispersive  powers  of  the  medium  ;    and  that 
as  glass.        therefore  by  varying  properly  the  ingredients  of  a  medium,  it  may  be  practicable,  without  greatly  varying  the 
total  refraction  and  dispersion,  still  to  produce  a  considerable  change  in  the  internal  arrangement  (if  we  may 
use  the  phrase)  of  the  spectrum  ;    and  therefore,    perhaps,  to   form  a  compound  medium  in  which  the  seven 
colours  shall  occupy  spaces  regulated  by  any  proposed  law,  (within  certain  limits.)     Now  if  a  medium  could  be 
so  compounded  as  to  have  the  same  scale  of  dispersions,  or  the  same  law  of  distribution  of  the  colours  as  crown 
glass  with  a  different  absolute  dispersion,  as  we  have  already  seen,  nothing  more  would  be  required  for  the  per- 
fection of  the  double  object-glass.     The  property  of  the  muriatic  acid  just  mentioned  puts  this  in  our  power. 
It  is  observed,  that  the  presence  of  a  metal  (antimony,  for  instance)  in  a  fluid,  while  it  gives  it  a  high  refrac- 
tive and  dispersive  power,  at  the  same  time  tends  to  dilate  the  more  refrangible   part  of  the  spectrum  beyond 
its  due  proportion   to  the  less.     On  the  other  hand,  the  presence  of  muriatic  acid  tends  to  produce  a  contrary 
effect,  contracting  the  more  refrangible  part  and  dilating  the  less,  beyond  that  proportion  which  they  have  in 
glass.     Hence,  Dr.  Blair  was  led  to  conclude,  that  by  mixing  muriatic  acid  with  metallic  solutions,  in  proportions 
to  be  determined  by  experience,  a  fluid  might  be  obtained  with  the  wished  for  property  ;  and  this  on  trial  he 
found  to  be  the  case.     The  metals  he  used  were  antimony  and  mercury ;  and  to  ensure  the  presence  of  a  suffi- 
cient quantity  of  muriatic  acid,  he  employed  them  in  the  state  of  muriates,  in  aqueous  solution  ;  or,  in  the  case 
of  mercury,  in  a  solution  of  sal  ammoniac,  which  is  a  compound  of  ammonia  and  muriatic  acid,  and  which   is 
capable  of  dissolving  a  considerably  greater  quantity  of  corrosive  sublimate  (muriate,  or  chloride   of  mercury) 
His  double   than  water  alone.     By  adding  liquid  muriatic  acid  to  the  compound  known  by  the  name  of  butter  of  antimony, 
object-          (chloride  of   antimony,)  or  sal   ammoniac  to  the   mercurial   solution,  he  succeeded  completely  in  obtaining  a 
!          spectrum  in  which  the  rays  followed  the  same  law  of  dispersion  as  in  crown  glass,  and  even  in  over-correcting 
'""•Tmema          secondary  spectrum,  so  as  to  place  its  exact  destruction  completely  in  his  power.     It  only  remained  to  form 
an  object-glass  on  these  principles.     Fig.  Ill  is  such  an  one,  in  which,  though  there  are  two  refractions  at  the 
confines  of  the  glass  and  fluid,  yet  the  chromatic  aberration,  as  Dr.  Blair  assures  us,  was  totally  destroyed,  and 
the  rays  of  different  colours  were  bent  from  their  rectilinear  course  with  the  same  equality  as  in  reflexion. 


LIGHT.  429 

Ij<m.  'fo  sucn  an  extent  has  Dr.  Blair  carried  these  interesting  experiments,  that  he  assures  us  he  has  found  it  prac-     P"'  !'• 

— ~v"~-'  ticable  to  construct  an  object-glass  of  nine  inches  focal  length,  capable  of  bearing  an  aperture  of  three  inches,  a  *— -V"" 
thing  which  assuredly  no  artist  would  ever  dream  of  attempting  with  glass  lenses;  and  we  cannot  close  this      477. 
account  of  his  labours  without  joining  in  a  wish   expressed  on  a  similar   occasion  by  Dr.  Brewster,   whose 
researches  on  dispersive  powers  have  so  worthily  filled  up  the  outline  sketched  by  his    predecessor,  that   this 
branch  of  practical   optics  may  be  resumed  with  the  attention  it  deserves,  by  artists  who  have  the  ready  means 
of  executing  the  experiments  it  would  require.    Could  solid  media  of  such  properties  be  discovered,  the  telescope 
would  become  a  new  instrument. 

These  experiments  of  Dr.  Blair  lead  to  the  remarkable  conclusion,  that  at  the  common  surface  of  two  media      478. 

a  white  ray  may  be  refracted  without  separation  into  its  coloured  elements.     In  fact,  /*  and  p!  being  the  refrac-  Case  of 

f  colourless 

tive  indices  of  the  media  for  any  ray  as  the  extreme  red,  will  be  their  relative  refractive  index  for  that  ray,  Sr, 

ii 
f    ,    »    /  common 

and  — —  will  be  the   relative  index  for  any  other  ray.     If,  then,  the  refractive  and  dispersive  powers  of'urhcejf 

ft  +  5  fa  .  two  media. 

the  media  be  such  that  —        ^     =   -^— ,  or  p.  S/=r  /  8/t,  that  is,  if  — — j  —  —  ;  and  if,  moreover,  this 

relation  hold  good  throughout  the  spectrum,  i.  e.  if  the  increments  of  the  refractive  indices,  in  proceeding  from 
the  red  to  the  violet  end  of  the  spectrum,  be  proportional  to  the  refractive  indices  themselves,  then  the  relative 
index  is  the  same  for  all  rays,  and  no  dispersion  will  take  place.  Now  this  gives  a  relation  between  the  disper- 
sive and  refractive  indices  of  the  two  media,  viz.  —  =  .  — -. =:  ^-  ;  and,  in  addition  to  this 

P  H          A»  -  1 

condition,  the  scale  of  dispersions  must  be  the  same  in  both  media.  According  as  the  dispersions  diifer  one  way 
or  the  other  from  this  precise  adjustment,  the  violet  ray  may  be  either  more  or  less  refracted  than  the  red  at  the 
common  surface  of  the  two  media. 

We  shall  terminate  the  theory  of  achromatic  object-glasses  with  a  problem  of  considerable  practical  import-      479. 
ance,  as  it  puts  it  in  our  power,  having  obtained  an  approximate  degree  of  achromaticity  in  an  object-glass,  to  Achromatic 
complete  the  destruction  of  the  colour  without   making  any  alteration  in  the  focal  lengths  or  curvatures  of  the  «hject-glass 

lenses,  by  merely  placing  them  at  a  greater  or  less  distance  from  one  another.  Mth  sepa- 

J  J  1  rated  lenses. 

Problem.  To  express  the  condition  of  achromaticity,  when  the  two  lenses  of  a  double  object-glass  are  placed 
at  a  distance  from  each  other,  (=  t.) 

Resuming  the  notation  of  Art.  251  and  268,  we  have 


and 


Now,  that  the  combination  may  be  achromatic,  we  must  have  S/iv  =  0  ;  and,  since  t  and  D  are  constant,  and 
L'  and  L''  only  vary  by  the  variations  of  /*',  /w,"  the  refractive  indices,  we  have   S  L'  =  (R'  —  R")  3/t'  = 

— p- — —  L'  =  p'L',  and  similarly  S  L"  =  p"  L",  so  that  substituting  we  get 

{l-*(L'+D)i-^.  Ji,-  =  0. 

Such  is  the  condition  of  achromaticity.     Since  it  depends  on  D,   it  appears  that  if  the  lenses  of  an  object-       430 
glass  be  not  close  together,  it  will  cease  to  be  achromatic  for  near  objects,  however  perfectly  the  colour  be  cor- 
rected for  distant  ones.     The  eye  therefore  cannot  be  achromatic  for  objects  at  all  distances,  its  lenses  being 
of  great  thickness  compared  to  their  focal  lengths ;  and,  therefore,  although  in  contact  at  their  adjacent  surfaces" 
yet  having  considerable  intervals  between  others. 

For  parallel  rays  the  equation  becomes  481. 

y/'L"(l-<L')*  =  -p'L'; 
hence,  the  dispersions  and  powers  of  the  lenses  being  given  their  interval  t  may  be  found  by  the  expression 

_!_/,_  */     P'     ^l 

L'    (       v      I?'  ~ur } 

The  condition  of  achromaticity,  were  the  lenses  placed  close  together,  would  be,  as  we  have  already  shown,       ^82. 


7 

?f" 

Li    -r-  U  ; 
3  IV'  -1-  - 

J"  —  L,"   + 

5/  '                  51 

•  n     i 

"7    = 
»L' 

:  «  Jj  ; 

^    +    ( 

1  —  f"W 

J       '        J  1   - 

!  CT.'  -1-  F 

(1  i  *   ' 

430 


L  I  G  H  T. 


^_^^_ t-^-  .   -J-TT  =  1 •     Hence,  whenever  this  fraction  is  less  than  unity,  that  is  whenever  L",  the  power  of  the 

concave  or  flint  lens  (which  we  here  suppose  to  be  the  second)  is  too  great ;  or  when,  as  the  opticians  call  it,  the 
colour  is  over-corrected,  the  object-glass  may  be  made  achromatic,  or  their  over-correction  remedied,  without  re- 
grinding  the  glasses,  merely  by  separating  the  lenses ;  for  in  this  case  the  quantity  under  the  radical  is  less  than 
unity,  and  therefore  t  is  positive,  a  condition  without  which  the  rays  could  not  be  refracted  as  we  have  supposed 
them. 

4gg  Moreover,  this  affords  a  practical  and  very  easy  means  of  ascertaining,  with  the  greatest  precision,  the  dis- 

persive ratio  of  the  two  media.  Let  a  convex  lens  of  crown  be  purposely  a  little  over-corrected  by  a 
concave  of  flint,  and  then  let  the  colour  be  destroyed  by  separating  the  lenses.  Measure  their  focal  lengths 

J  and  the  interval  t  between  them  in  this  state,  and  we  have  at  once  for  the  value  of  ro  the 


and 


L" 
dispersive  ratio, 


L" 


p" 


(1-tL1)'' 


Part  H. 


§  III.  Of  the  Absorption  or  Extinction  of  Light  by  uncrystallized  Media. 


484. 

All  media 
absorb  light. 


485. 


486. 

And  all 
absorb  the 
different 
colours 
unequally. 


487. 
Experiment 


Transparency  is  the  quality  by  which  media  allow  rays  of  light  freely  to  pass  through  their  substance,  or,  it 
may  be,  between  their  molecules  ;  and  is  said  to  be  more  or  less  perfect,  according  as  a  more  or  less  consider- 
able part  of  the  whole  light  which  enters  them  finds  its  way  through.  Among  media,  consisting  of  ponderable 
matter,  we  know  of  none  whose  transparency  is  perfect.  Whether  it  be  that  some  of  the  rays  in  their  passage 
encounter  bodily  the  molecules  of  the  media,  and  are  thereby  reflected  ;  or,  if  this  supposition  be  thought  too 
coarse  and  unrefined  for  the  present  state  of  science,  be  stopped  or  turned  aside  by  the  forces  which  reside  in 
the  ultimate  atoms  of  bodies,  without  actual  encounter,  or  otherwise  detained  or  neutralized  by  them  ;  certain  it 
is,  that  even  in  the  most  rare  and  transparent  media,  such  as  air,  water,  and  glass,  a  beam  of  light  intromittecl, 
is  gradually  extinguished,  and  becomes  more  and  more  feeble  as  it  penetrates  to  a  greater  depth  within  them, 
and  ultimately  becomes  too  faint  to  affect  our  organs.  Thus,  at  the  tops  of  very  high  mountains,  a  much 
greater  multitude  of  stars  is  visible  to  the  naked  eye  than  on  the  plains  at  their  feet ;  the  weak  light  of  the 
smallest  of  them  being  too  much  reduced  in  its  passage  through  the  lower  atmospheric  strata  to  affect  the  sight. 
Thus,  too,  objects  cease  to  be  visible  at  great  depths  below  water,  however  free  from  visible  impurities,  &c.  Dr. 
Olbers  has  even  supposed  the  same  to  hold  good  with  the  imponderable  media  (if  any)  of  the  celestial  spaces, 
and  conceives  this  to  be  the  cause  why  so  few  stars  (not  more  than  about  five  or  ten  millions)  can  be  seen  with 
the  most  powerful  telescopes.  It  is  probable  that  we  shall  be  long  without  means  of  confirming  or  refuting 
this  singular  doctrine. 

On  the  other  hand,  though  no  body  in  nature  be  perfectly,  all  are  to  a  certain  degree,  transparent.  One  of 
the  densest  of  metals,  gold,  may  actually  be  beaten  so  thin  as  to  allow  light  to  pass  through  it ;  and  that  it  passes 
through  the  substance  of  the  metal,  not  through  cracks  or  holes  too  small  to  be  detected  by  the  eye,  is  evident 
from  the  colour  of  the  transmitted  light,  which  is  green,  even  when  the  incident  light  is  white.  The  most 
opaque  of  bodies,  charcoal,  in  a  different  state  of  aggregation,  (as  diamond,)  is  one  of  the  most  perfectly  trans- 
parent ;  and  all  coloured  bodies,  however  deep  their  hues,  and  however  seemingly  opaque,  must  necessarily  be 
rendered  visible  by  rays  which  have  entered  their  substance  ;  for  if  reflected  at  their  surfaces,  they  would  all 
appear  white  alike.  Were  the  colours  of  bodies  strictly  superficial,  no  variation  in  their  thickness  could  affect 
their  hue  ;  but,  so  far  is  this  from  being  the  case,  that  all  coloured  bodies,  however  intense  their  tint,  become 
paler  by  diminution  of  thickness.  Thus  the  powders  of  all  coloured  bodies,  or  the  streak  they  leave  when 
rubbed  on  substances  harder  than  themselves,  have  much  paler  colours  than  the  same  bodies  in  mass. 

This  gradual  diminution  in  the  intensity  of  a  transmitted  ray  in  its  progress  through  imperfectly  transparent 
media,  is  termed  its  absorption.  It  is  never  found  to  affect  equally  rays  of  all  colours,  some  being  always  absorbed 
in  preference  to  others ;  and  it  is  on  this  preference  that  the  colours  of  all  such  media,  as  seen  by  transmitted 
light,  depend.  A  white  ray  transmitted  through  a  perfectly  transparent  medium,  ought  to  contain  at  its  emer- 
gence the  same  proportional  quantity  of  all  the  coloured  rays,  because  the  part  reflected  at  its  anterior  and 
posterior  surfaces  is  colourless ;  but,  in  point  of  fact,  such  perfect  want  of  colour  in  the  transmitted  beam  is 
never  observed.  Media,  then,  are  unequally  transparent  for  the  differently  coloured  rays.  Each  ray  of  the 
spectrum  has,  for  every  different  medium  in  nature,  its  own  peculiar  index  of  transparency,  just  as  the  index  of 
refraction  differs  for  different  rays  and  different  media. 

The  most  striking  way  in  which  this  different  absorptive  power  of  one  and  the  same  medium  on  differently 
coloured  rays  can  be  exhibited,  is  to  look  through  a  plain  and  polished  piece  of  smalt-blue  glass,  (a  rich  deep  blue, 
very  common  in  the  arts — such  as  sugar-basins,  finger-glasses,  &c.  are  often  made  of,)  at  the  image  of  any  narrow 
line  of  light  (as  the  crack  in  a  window-shutter  of  a  darkened  room)  refracted  through  a  prism  whose  edge  is 
parallel  to  the  line,  and  placed  in  its  situation  of  minimum  deviation.  If  the  glass  be  extremely  thin,  all  the 
colours  are  seen  ;  but  if  of  moderate  thickness  (as  VT  inch)  the  spectrum  will  put  on  a  very  singular  and  striking 
appearance.  It  will  appear  composed  of  several  detached  portions  separated  by  broad  and  perfectly  black 


L  I  G  H  T  4ol 

intervals,  the  rays  which  correspond  to  those  points  in  the  perfect  spectrum  being  entirely  extinguished.    If  a  less     Part  II. 
thickness  be  employed,  the  intervals,  instead  of  being  perfectly  dark,  are  feebly  and  irregularly  illuminated,  some  ^— - v"~' 
parts  of  them  being  less  enfeebled  than  others.     If  the  thickness,  on  the  other  hand,  be  increased,  the   black 
spaces  become  broader,  till  at  length  all  the  colours  intermediate  between  the  extreme  red  and  extreme  violet  are 
totally  destroyed. 

The  simplest  hypothesis  we  can  form  of  the  extinction  of  a  beam  of  homogeneous  light  in  passing  through  a  488. 
homogeneous  medium,  is,  that  for  every  equal  thickness  of  the  medium  passed  through,  an  equal  aliquot  part  of  Law  of  . 
the  rays,  which,  up  to  that  depth  had  escaped  absorption,  is  extinguished.  Thus,  if  1000  Ted  rays  fall  on  and  ra 
enter  into  a  certain  green  glass,  and  if  100  be  extinguished  in  traversing  the  first  tenth  of  an  inch,  there  will 
remain  900  which  have  penetrated  so  far;  and  of  these  one-tenth,  or  90,  will  be  extinguished  in  the  next  tenth 
of  an  inch,  leaving  810,  out  of  which  again  a  tenth,  or  81,  will  be  extinguished  in  traversing  the  third  tenth, 
leaving  729,  and  so  on.  In  other  words,  the  quantity  unabsorbed,  after  the  beam  has  traversed  any  thickness  of 
the  medium,  will  diminish  in  geometrical  progression,  as  t  increases  in  arithmetical.  So  that  if  1  be  taken  for 
the  whole  number  of  intromitted  rays,  and  y  for  the  number  that  escape  absorption  in  traversing  an  unit  of 
thickness,  y '  will  represent  the  number  escaping,  after  traversing  any  other  thickness,  =  t.  This  only  supposes 
that  the  rays  in  the  act  of  traversing  one  stratum  of  a  medium  acquire  no  additional  facility  to  penetrate  the 
remainder.  In  this  doctrine,  y  is  necessarily  a  fraction  smaller  than  unity,  and  depending  on  the  nature  both  of 
the  ray  and  the  medium.  Hence,  if  C  represent  the  number  of  equally  illuminating  rays  of  the  extreme  red  in 
a  beam  of  white  light,  C'  that  of  the  next  degree  of  refran^ibility,  and  so  on  ;  the  beam  of  white  light  will  be 
represented  by  C  +  C'  +  C"  +  &c. ;  and  the  transmitted  beam,  after  traversing  the  thickness  t,  will  be  properly 
expressed  by 

C.y'  +  C'.y"  +  C".y"'  +  &c. 

Each  term  representing  the  intensity  of  the  particular  ray  to  which  it  corresponds,  or  its  ratio  to  what  it  is  in  the 
original  white  beam. 

It  is  evident  from  this,  that,  strictly  speaking,  total  extinction  can  never  take  place  by  any  finite  thickness  of      489. 
the  medium  ;  but  if  the  fraction  y  for  any  ray  be  at  all  small,  a  moderate  increase  in  the  thickness,  (which  enters 
as  an  exponent,)  will  reduce  the  fraction  y'  to  a,  quantity  perfectly  insensible.     Thus,  in  the  case  taken  above, 
where  a  tenth  of  an  inch  of  green  glass  destroys  one-tenth  only  of  the  red  rays,  a  whole  inch  will  allow  to  pass 

(9  V° 
— — -J    ,  or  304  rays   out  of  a  thousand,   while  ten   times  that  thickness,  or  10  inches,  will  suffer  only 

/    9  V°° 

l~7jj- J      =  0.0000266,  or  less  than  three  rays  out  of  100,000  to  pass,   which   amounts  to    almost   absolute 

opacity. 

If  x  be  the  index  of  refraction  of  any  ray  in  the  water  spectrum,  we  may  regard  y  as  a  function  of  x ;  and  if  on       490. 
the  line  RV,  (fig.  11 2,)  representing  the  whole  length  of  the  water  spectrum,  we  erect  ordinates,  Rr,  MN,V  Vequal  L 
to  unity  and  to  each  other ;  and  also  other  ordinates  R  r,  M  P,  V  u  representing  the  values  of  y  for  the  rays  at  j^rpm°ediu 
the  corresponding  points;  the  curve  rPv,  the  locus  of  P,  will  be,  as  it  were,  a  type,  or  geometrical  picture  of  expressed 
the   action  of   the  medium  on   the  spectrum,  and  the  straight  line  RN  V  will  be  a  similar  type  of  a  perfectly  bya  curve, 
transparent  medium.     Now  if  this  be  supposed  the  case  when  the  thickness  of  the  medium  is  1,  if  we  take  F|g-  H&- 
always  M  f  :  M  P  :  :  M  P  :  M  N,  and  M  P" :  M  P' : :  M  P' :  M  P,  &c.  and  so  on,  the  loci  of  P'  P",  &c.  will  be 
curves  representing  the  quantities  of  the  rays  transmitted  by  the  thicknesses  2,  3,  &c.  of  the  medium,  and  so  for 
intermediate  thicknesses,  or  for  a  thickness  less  than  1,  as  in  the  curve  5  irv. 

Hence,  whatever  be  the  colour  of  a  medium,  if  its  thickness  be  infinitely  diminished,  it  will  transmit  all  the       491 
rays  indifferently ;  for  when  t  =  0,  y '  =  1,  whatever  be  y  ;  and  the  curve  f>  v  v  approaches  infinitely  near  to  the 
line  R'N  V.     Thus  all  coloured  glasses  blown  into  excessively  thin  bubbles  are  colourless,  and  so  is  the  foam 
of  coloured  liquids. 

Again,  if  there  be  any,  the  least,  preference  given  by  the  medium  to  the  transmission  of  certain  rays  beyond       492. 
others,  the  thickness  of  the  medium  may  be  so  far  increased  as  to  give  it  any  assignable  depth  of  tint ;  for  if  y 
be  ever  so  little  less  than  unity,  and  if  between  the  values  of  y  for  different  rays  there  be  ever  so  little  difference, 
t  may  be  so  increased  as  to  make  y '  as  small  as  we  please,  and  the  ratio  of  y '  to  y' '  as  different  from  unity  as 
we  please. 

In  very  deep  coloured  media  all  the  values  of  y  are  small.     If  they  were   equal,  the  medium  would  merely       493. 
stop  light,  without  colouring  the  transmitted  beam,  but  no  such  media  are  at  present  known. 

If  the  curve  rPv,  or  the  type  of  an  absorbent  medium  have  a  maximum  in  any  part  of  the  spectrum,  as  in  the       4 
green,  for  instance,  (fig.  113  ;)  then,  whatever  be  the  proportion  in  which  the  other  rays  enter,  by  a  sufficient  J^™* ^ 
increase  of  thickness,  that  colour  will  be  rendered  predominant;  and  the  ultimate  tint  of  the  medium,  or  the  gbsorpt\\e 
last  ray  it  is  capable  of  transmitting,  will  be  a  pure  homogeneous  light  of  that  particular  refrangibility  to  which  medium. 
the  maximum  ordinate  corresponds.      Thus  green   glasses,  by  an  increase  of  thickness,  become  greener  and  Fig.  113. 
greener,  their  type  being  as  in  fig.  113;  while  yellow  ones,  whose  type  is  as   in  fig.  114,  change  their  tint  by 
reduplication,  and  pass  through  brown  to  red. 

This  change  of  tint  by  increase  of  thickness  is  no  uncommon  phenomenon  ;  and  though  at  first  sight  para-      495. 
doxical,  yet  is  a  necessary  consequence  of  the  doctrine  here  laid  down.     If  we  enclose  a  pretty  strong  solution  Tint 
of  sap-green,  or,  still  better,  of  muriate  of  chromium  in  a  thin  hollow  glass  wedge,  and  if  we  look  through  the  Ranges  by 
edge  where  it  is  thinnest,  at  white  paper,  or  at  the  white  light  of  the  clouds,  it  appears  of  a  fine  green;  but  'fJfJk'iknraj0* 
we  slide  the  wedge  before  the  eye  gradually  so  as  to  look  successively  through  a  greater  and  greater  thickness 


432 


LIGHT. 


Case  of 
green-red 
medium. 
Fig.  115. 


Light,  of  the  liquid,  the  green  tint  grows  livid,  and  passes  through  a  sort  of  neutral,  brownish  hue,  to  a  deep  blood- 
red.  To  understand  this,  we  must  observe,  that  the  curves  expressing  the  types  of  different  absorbent  media  ' 
admit  the  most  capricious  variety  of  form,  and  very  frequently  have  several  maxima  and  minima  corresponding 
to  as  many  different  colours.  The  green  liquids  in  question  have  two  distinct  maxima,  as  in  fig.  115  ;  the  one 
corresponding  to  the  extreme  red,  the  other  to  the  green,  but  the  absolute  lengths  of  the  maximum  ordinates 
are  unequal,  the  red  being  the  greater.  But  as  the  extreme  red  is  a  very  feebly  illuminating  ray,  while  on  the 
other  hand  the  green  is  vivid,  and  affects  the  eye  powerfully,  the  latter  at  first  predominates  over  the  former,  and 
entirely  prevents  its  becoming  sensible  ;  and  it  is  not  till  the  thickness  is  so  far  increased  as  to  leave  a  very  great 
preponderance  of  those  obscure  red  rays,  and  subdue  their  rivals,  as  in  the  case  represented  by  the  lowest  of 
the  dotted  curves  in  the  figure,  that  we  become  sensible  of  their  influence  on  the  tint.  Suppose,  for  instance, 
Numerical  to  illustrate  this  by  a  numerical  example,  the  index  of  transparency,  or  value  of  y,  in  muriate  of  chromium,  to 
illustration.  De  for  extreme  red  rays,  0.9  ;  for  the  mean  red,  orange,  and  yellow,  0.1 ;  for  green,  0.5  ;  and  for  blue,  indigo, 
and  violet,  0.1  each;  and  suppose,  moreover,  in  a  beam  of  white  light,  consisting  of  10,000  rays,  all  equally 
illuminative,  the  proportions  corresponding  to  the  different  colours  to  be  as  follows  : 


Part  II. 


Extreme  red. 
200 


Red  and  orange. 
1300 


Yellow. 
3000 


Green. 
2800 


Blue. 
1200 


Indigo. 
1000 


Violet. 
500. 


Then,  after  passing  through  a  thickness  equal  to  1  of  the  medium,  the  proportions  in  the  transmitted  beam 
would  be 


Extreme  red. 
180 


Red  and  orange. 
{30 


Yellow. 
300 


Green. 

1400 


After  traversing  a  second  unit  of  thickness,  they  would  be 


Extreme  red. 
162 

and  after  a  third,  a 

Extreme  red 
146 
131 
118 
106 

Red  and  orange. 
13 

fourth,  a  fifth,  and  sixth 

Red  and  orange. 
1 
0 
0 
0 

Yellow. 
30 

respectively, 

Yellow. 
3 
0 
0 
0 

Green. 
700 

Green. 
350 
175 
87 
43 

Blue. 

120 


Blue 

12 


Blue. 
1 
0 
0 
0 


Indigo. 

100 


Indigo. 
10 


Indigo. 

i 
o 

0 
0 


Violet. 
50. 


Violet. 
5. 


Violet. 
0 
0 

(I 

0. 


496. 

Relative  il- 
luminative 
power  of 
the  several 
prismatic 
rays. 
Fig    116. 


497 


Fig.  117. 


Thus  we  see,  that  in  the  first  of  these  transmitted  beams  the  green  greatly  preponderates ,  after  the  second 
transmission,  it  is  still  the  distinguishing  colour ;  but  after  the  third,  the  red  bears  a  proportion  to  it  large 
enough  to  impair  materially  the  purity  of  its  tint.  The  fourth  transmission  may  be  regarded  as  totally  extin- 
guishing all  the  other  colours,  and  leaving  a  neutral  tint  between  red  and  green  ;  while,  in  all  the  tints 
produced  by  further  successive  transmissions,  the  red  preponderates  continually  more  and  more,  till  at  length 
the  tint  becomes  no  way  distinguishable  from  the  homogeneous  red  of  the  extremity  of  the  spectrum. 

Whether  we  suppose  the  obscurer  parts  of  the  spectrum  to  consist  of  fewer  rays  equally  illuminative,  or  of 
the  same  number  of  rays  of  less  intrinsic  illuminating  power  with  the  brighter,  obviously  makes  no  difference  in 
the  conclusion,  but  the  former  supposition  has  the  advantage  of  affording  a  hold  to  numerical  estimation  which 
the  latter  does  not.  In  the  instance  here  taken,  the  numbers  are  assumed  at  random.  But  Fraunhofer  has  made  a 
series  of  experiments  expressly  to  determine  numerically  the  illuminating  power  of  the  different  rays  of  the  spectrum. 
According  to  which,  he  has  constructed  the  curve  fig.  116,  whose  ordinate  represents  the  illuminative  power  of 
the  ray  in  that  part  of  the  spectrum  on  which  it  is  supposed  erected,  or  the  proportional  number  of  equally 
illuminative  rays  of  that  refrangibility  in  white  light.  If  we  would  take  this  into  consideration  in  our  geome- 
trical construction,  we  must  suppose  the  type  of  white  light,  instead  of  being  a  straight  line,  as  in  fig.  1 12.  ... 
114,  to  be  a  curve  similar  to  fig.  116,  and  the  other  derivative  curves  to  be  derived  from  it  by  the  same  rules 
as  above.  But  as  the  only  use  of  such  representations  is  to  express  concisely  to  the  eye  the  general  scale  of 
action  of  a  medium  on  the  spectrum,  this  is  rather  a  disadvantageous  than  a  useful  refinement. 

To  take  another  instance.  If  we  examine  various  thicknesses  of  the  smalt-blue  glass  above  noticed,  it  will 
be  found  to  appear  purely  blue  in  small  thicknesses.  As  the  thickness  increases,  a  purple  tinge  comes  on,  which 
becomes  more  and  more  ruddy,  and  finally  passes  to  a  deep  red  ;  a  great  thickness  being,  however,  required 
to  produce  this  effect.  If  we  examine  the  tints  by  a  prism,  we  shall  find  the  type  of  this  medium  to  be  as  in 
fig.  117,  having  four  maximum  ordinates,  the  greatest  corresponding  to  a  ray  at  the  very  farthest  extremity  of  the 
red,  and  diminishing  with  such  rapidity  as  to  cause  an  almost  perfect  insulation  of  this  ray ;  the  next  corresponds 
to  a  red  of  mean  refrangibility,  the  next  to  the  mean  yellow,  and  the  last  to  the  violet,  the  ordinate  increasing 
continually  to  the  end  of  the  spectrum.  Thus,  when  a  piece  of  such  glass  of  the  thickness  0.042  inch  was  used, 
the  red  portion  of  the  spectrum  was  separated  into  two,  the  least  refracted  being  a  well  defined  band  of  per 
fectly  homogeneous  and  purely  red  light,  separated  from  the  other  red  by  a  band  of  considerable  breadth,  and 
totally  black.  This  red  was  nearly  homogeneous ;  its  tint,  however,  differing  in  no  respect  from  the  former, 
and  being  free  from  the  slightest  shade  of  orange.  Its  most  refracted  limit  came  very  nearly  up  to  the  dark  line 
D  in  the  spectrum.  A  small,  sharp, black  line  separated  this  red  from  the  yellow,  which  was  a  pretty  well  defined 
band  of  great  brilliancy  and  purity  of  colour,  of  a  breadth  exceeding  that  of  the  first  red,  and  bounded  on  the 


LIGHT.  433 

preen  side  by  an  obscure  but  not  quite  black  interval.     The  green  was  dull  and  ill  defined,  but  the  violet  was     Part  II. 
'  transmitted  witli  very  little  loss.     A  double  thickness  (0.084  inch)  obliterated  the  second  red,  greatly  enfeebled  "— ~v~— • 
the  yellow,  leaving  it  now  sharply  divided  from  the  green,  which  was  also  extremely  enfeebled.     The  extreme 
red,  however,   retained  nearly  its  whole  light,  and  the  violet  was  very  little  weakened.     When  a  great  many 
thicknesses  were  laid  together,  the  extreme  red  and  extreme  violet  only  passed. 

Among  transparent  media  of  most  ordinary  occurrence,  we  may  distinguish,  first,  those  whose  type  has  its       498. 
ordinate  decreasing  regularly,  with  more  or  less  rapidity  from  the  red  to  the  violet  end  of  the  spectrum,  or  Red  media, 
which  absorb  the  rays  with  an  energy  more  or  less  nearly  in  some  direct  ratio  of  their  refrangibility.     In  red 
and  scarlet  media  the  absorbent  power  increases  very  rapidly,  as  we  proceed  from  the  red  to  the  violet.     In 
yellow,  orange,  and  brown  ones,  less  so  ;  but  all  of  them  act  with  great  energy  on  the  violet  rays,  and  produce 
a  total  obliteration  of  them.     In  consequence  of  this,  by  an  increase  of  thickness,  all  these  media  finally  become 
red.     Examples :    red,  scarlet,  brown,  and  yellow  glasses ;   port  wine,  infusion  of  saffron,  permuriate  of  iron, 
muriate  of  gold,  brandy,  India  soy,  &c. 

Among  green  media,  the  generality  have  a  single  maximum  of  transmission  corresponding  to  some  part  of      499. 
the  green  rays,  and  their  hue  in   consequence  becomes  more  purely  green  by  increase  of  thickness.     Of  this  Simple 
kind  are  green  glasses,  green  solutions  of  copper,  nickel,  &c.     They  absorb  both  ends  of  the  spectrum  with  green  media, 
great  energy  ;  the  red,  however,  more  so,  if  the  tint  verges  to  blue  ;    the  violet,  if  to  yellow.     Besides  these, 
however,  are  to  be   remarked  media  in  which  the  type  has  two  maxima ;    such  may  be   termed  dichromatic,  Dichromatic 
having  really  two  distinct  colours.     In  most  of  these,  the  green  maximum  is  less  than  the  red ;  and  the  green  media. 
tint,  in  consequence,  loses  purity  by  increase  of  thickness,  and  passes  through  a  livid  neutral  hue  to  red,  though 
this  is  not  always  the  case.     Examples  :    muriate  of  chrome,  solution  of  sap-green,  manganesiate  of  potash, 
alkaline  infusion  of  the  petals  of  the  peonia  ollicinalis  and  many  other  red  flowers,  and  mixtures  of  red  and 
blue  or  green  media. 

Blue  media  admit  of  great  variety,  and  are  generally  dichromatic,  having  two  or  even  a  great  many  maxima       500 
and  minima  in  their  types ;  but  their  distinguishing  character  is  a  powerful  absorption  of  the  more  luminous  Blue  media. 
red  rays  and  the  green,  and  a  feeble  action  on  the  more  refrangible  part  of  the   spectrum.     Among  those  whose 
energy  of  absorption  appears  to  increase  regularly  and  rapidly  from  the  violet  to  the  red  end  of  the  spectrum, 
we  may  place  the  blue  solutions  of  copper.     The  best  example  is  the  magnificent  blue  liquid  formed  by  super- 
saturating sulphate  of  copper  with  carbonate  of  ammonia.     The  extreme  violet  ray  seems  capable,  of  passing 
through  almost  any  thickness  of  this  medium ;  and  this  property,  joined  to  the  unalterable  nature  of  the  solution, 
and  the  facility  of  its  preparation,  render  it  of  great  value  in  optical  researches.     A  vessel,  or  tube,  of  some  Insulation  of 
inches  in  length,  closed  at  two  ends  with  glass  plates,  and  filled  with  this  liquid,  is  the  best  resource  for  experi-  ".le  pxtreme 
ments  on  the  violet  rays.     Ammonio-oxalate  of  nickel  transmits  the  blue  and  extreme  red,  but  stops  the  violet. 

Purple  media  act  by  absorbing  the  middle  of  the  spectrum,  and  are  therefore  necessarily  always  dichromatic,  5^1 
some  of  them  having  red  and  others  violet  for  their  ultimate  or  terminal  tint.  Example:  solution  of  archil  ;  purplc 
purple,  plum-coloured,  and  crimson  glasses  ;  acid  and  alkaline  solutions  of  cobalt,  &c.  They  may  be  termed  red-  media. 
purple  and  violet-purple,  according  to  their  terminal  tint. 

In  combinations  of  media,  the  ray  finally  transmitted  is  the  residuum  of  the  action  of  each.     If  x,  y,  z  be      503. 
the  indices  of  transmissibility  of  a  given  ray  C  in  the  spectrum  for  the  several  media,  and  r,  s,  t  their  thicknesses,  Combina- 
the  transmitted  portion  of  this  ray  will  be  C  .  xr  ys  z' ;  and  the  residuum  of  a  beam  of  white  light  (supposing  'ions  of 
none  lost  by  reflexion  at  the  surfaces)  after  undergoing  the  absorptive  action  of  all  the  media,  will  be  media. 

C  .  xr  y'z'  +  C'.  xlr  y"  z"  +  &c, 

An  expression  which  shows  that  it  is  indifferent  in  what  order  the  media  are  placed.  They  may  therefore  be 
mixed,  unless  a  chemical  action  take  place.  Thus  also,  by  the  same  construction  as  that  by  which  the  type  I 
of  the  first  medium  is  derived  from  the  straight  line  representing  white  light,  may  another  type  2  be  derived  from 
1,  and  so  on ;  and  thus  an  endless  variety  of  types  will  originate,  having  so  many  tints  corresponding  to  them. 

This   circumstance  enables  us   to  insulate,  in  a  state  of  considerable  homogeneity,  various  rays.     Thus,  by       303. 
combining  with  the  smalt-blue  glass,  already  mentioned,  any  brown  or  red  glass  of  tolerable  fulness  and  purity  Insulation 
of  colour,  a  combination   will   be  formed  absolutely  impermeable  to   any  but  the  extreme  red   ray,  and  the 
refrangibility  of  this  is  so  strictly  definite  as  to  allow  of  its  being  used  as  a  standard  ray  in  all  optical  inquiries,  geneous  rej 
which  is  the  more  valuable,  as  the  coloured  glasses  by  which  it  is  insulated  are  the  most  common  of  any  which  ray. 
occur  in  the  shops,  and  may  be  had  at  any  glazier's.     If  to  such  a  combination  a  green  glass  be  added,  a  total 
stoppage  of  all  light  takes  place.     The  same  kind  of  glass,  too,  enables  us  to  insulate  the  yellow  ray,  corres-  Insulation 
ponding  to  the  maximum  Y  in  the  type  fig.  117,  by  combining  it  with  a  brown  glass  to  stop  out  the  more,  and  ™J** 
a  green  to  destroy  the  less,  refrangible  rays,  and  by  their  means  the  existence   of  a  considerable  breadth  of" 
yellow  light,  evidently  not  depending  on  a  mixture,  or  mutual  encroachment  of  red  and  green,  may  be  exhibited 
in  the  solar  spectrum. 

It  has  been  found  by  Dr.  Brewster,  that  the  proportions  of  the  different  coloured  rays  absorbed  by  media       3®  * 
depend  on  their  temperature.     The  tints  of  bodies  generally  deepen  by  the  application  of  heat,  as  is  known  to  'V^™','""  °' 
all  who  are  familiar  with  the  use  of  the  blow-pipe ;  thus  minium  and  red  oxide  of  mercury  deepen  in  their  hues  ^0^  „* 
by  heat  till  they  become  almost  black,  but  recover  their  red  colours  on  cooling.     Dr.  Brewster  has,  however,  heat, 
produced  instances,  not  merely  among  artificial  glasses,  but  among  transparent  minerals,  where  a  transition  takes 
place  from  red  to  green  on  the  application  of  a  high  temperature ;    the  original  tint  being,  however,  restated  on 
cooling,  and  no  chemical  alteration  having  been  produced  in  the  medium. 

The  analysis  of  the  spectrum  by  coloured  media  presents  several  circumstances  worthy  of  remark.     First,  the        50;> 
irregular  and  singular  distribution  in  the  dark  bands  which  cross  the  spectrum,  when  viewed  through    such 
VOL.  iv.  3  i. 


434  LIGHT. 

Light-      media  as  have  several  maxima  of  transmission,  obviously  leads  us  to  refer  Fraunhofer's  Fixed  lines,  and  the     Pa»  1J- 
s"~v"^p/  analogous  phenomena  to  be  noticed  in  the  light  from  other  sources,  to  the  same  cause,  whatever  it  may  be,  v"lp"v-^- 
which  determines  the  absorption  of  some  ray  in  preference  to  others.     It  is  no  impossible  supposition,  that  the 
deficient  rays  in  the  light  of  the  sun  and  stars  may  be  absorbed  in  passing  through  their  own  atmospheres,  or,  to 
approach  still  nearer  to  the  origin  of  the  light,  we  may  conceive  a  ray  stifled  in  the  very  act  of  emanation  from 
a  luminous  molecule  by  an  intense  absorbent  power  residing  in  the  molecule  itself;    or,  in  a  word,  the  same 
indisposition  in  the  molecules  of  an  absorbent   body  to  permit  the  propagation  of  any  particular  coloured  ray 
through,  or  near  them,  may  constitute  an  obstacle  in  limine  to  the  production  of  the  ray  from    them.     At  all 
events,  the  phenomena  are  obviously  related,  though  we  may  not  yet  be  able  to  trace  the  particular  nature   of 
their  connection. 

506.  The  next  circumstance  to  be  observed  is,  that  when  examined  through  absorbent  media  all  idea  of  regular 
gradation  of  colour  from  one  end  to  the  other  of  the  spectrum  is  destroyed.     Rays' of  widely  different  refrangi- 
bility,  as  the  two  reds  noticed  in  Art.  497,  have  absolutely  the  same  colour,  and  cannot  be  distinguished.     On 
the  other  hand,  the  transition  from  pure  red  to  pure  yellow,  in  the  case  there  described,  is  quite  sudden,  and  the 
contrast  of  colours  most  striking,  while  the  dark  interval  which  separates  them,  by  properly  adjusting  the 
thickness  of  the  glass,  may  be  rendered  very  small  without  any  tinge  of  orange  becoming  perceptible.     What 
then,  we  may  ask,  is  become  of  the  orange ;  and  how  is  it,  that  its  place  is  partly  supplied  with  red  on  one  side, 
and  yellow  on  the  other  ?    These  phenomena  certainly  lead  us  very  strongly  to  believe  that  the  analysis  of  white 
light  by  the  prism  is  not  the  only  analysis  of  which  it  admits,  and  that  the  connection  between  the  refrangibility 
and  colour  of  a  ray  is  not  so  absolute  as  Newton  supposed.     Colour  is  a  sensation  excited  by  the  rays  of  light, 
and  since  two  rays  of  different  refrangibilities  are  found  to  excite  absolutely  the  same  sensation  of  colour,  there 
is  no  primd  facie  absurdity  in  supposing  the  converse, — that  two  rays  capable  of  exciting  sensations  of  different 
colours  may  have  identical  indices  of  refraclion.     It  is  evident,  that  if  this  be  the  case,  no  mere  change  of 
direction  by  refractions  through  prisms,  &c.  could  ever  separate  them ;  but  should  they  be  differently  absorbable 
by  a  medium  through  which  they  pass,  an  analysis  of  the  compound  ray  would  take  place  by  the  destruction  of 
one  of  its  parts.     This  idea  has    been   advocated  by  Dr.  Brewster,  in  a  Paper   published  in  the  Edinburgh 
Philosophical  Transactions,  vol.   ix.,  and  the  same  consequence  appears  to  follow  from  other  experiments,  pub- 
lished in  the  same  volume  of  that  collection.     According  to  this  doctrine,  the  spectrum  would  consist  of  at  least 
three  distinct  spectra  of  different  colours,  red,  yellow,  and  blue,  over-lapping  each    other,  and   each   having  a 
maximum  of  intensity  at  those  points  where  the  compound  spectrum  has  the  strongest  and  brightest  tint  of 
that  colour. 

507.  It  must  be  confessed,  however,  that  this  doctrine  is  not  without  its  objections  ;  one  of  the  most  formidable  of 
Cases  of       which  may  be  drawn  from  the  curious  affection  of  vision  occasionally  (and  not  very  rarely)  met  with  in  certain 
persons  who  individuals,  who  distinguish  only  two  colours,  which  (when  carefully  questioned  and  examined  by  presenting  to 
coTou'rs''  tW°  tnem>  not  tne  ordinary  compound  colours  of  painters,   but  optical  tints  of  known  composition)  are  generally 

found  to  be  yellow  and  blue.  We  have  examined  with  some  attention  a  very  eminent  optician,  whose  eyes  (or 
rather  eye,  having  lost  the  sight  of  one  by  an  accident)  have  this  curious  peculiarity,  and  have  satisfied  ourselves, 
contrary  to  the  received  opinion,  that  all  the  prismatic  rays  have  the  power  of  exciting  and  affecting  them  with 
the  sensation  of  light,  and  producing  distinct  vision,  so  that  the  defect  arises  from  no  insensibility  of  the  retina 
to  rays  of  any  particular  refrangibility,  nor  to  any  colouring  matter  in  the  humours  of  the  eye,  preventing 
certain  rays  from  reaching  the  retina,  (as  has  been  ingeniously  supposed,)  but  from  a  defect  in  the  sensorium, 
by  which  it  is  rendered  incapable  of  appreciating  exactly  those  differences  between  rays  on  which  their  colour 
depends.  The  following  is  the  result  of  a  series  of  trials,  in  which  a  succession  of  optical  tints  produced  by 
polarized  light,  passing  through  an  inclined  plate  of  mica,  in  a  manner  hereafter  to  be  described,  was  submitted 
to  his  judgment.  In  each  case,  two  uniformly  coloured  circular  spaces  placed  side  by  side,  and  having  comple- 
mentary tints  (i.  e.  such  that  the  sum  of  their  light  shall  be  white)  were  presented,  and  the  result  of  his  judgment 
is  here  given  in  his  own  words. 


LIGHT. 


435 


Light. 


Colours  according  to  the  judgment  of  an  ordinary  eye. 

Colours  as  named  by  the  individual  in  question. 

Inclination 
of  the 
plate  of 
mica  to  eye. 

Circle  to  the  left. 

Circle  to  the  right. 

Circle  to  the  left. 

Circle  to  the  right. 

Pale  green. 

Pale  pink. 

Both   alike,  no  more  colour 

in  them  than  in  the  cloudy 

89.5 

sky  out  of  window. 

Dirty  white. 

Ditto,  both  alike. 

Both  darker  than  before,  but 

no  colour. 

85.0 

Fine  bright  pink. 

Fine  green,  a  little  verging 

Very  pale  tinge  of  blue. 

Very  pale  tinge  of  blue. 

81.1 

on  bluish. 

White. 

White. 

Yellow. 

Blue. 

76.3 

The  limit  of 

pink  and  red. 

Both  more  coloured 

than  before 

Rich  grass  green. 

Rich  crimson. 

Yellow. 

Blue. 

74.9 

Better,  but  neither 

full  colours. 

Dull  greenish  blue. 

Pale  brick  red. 

Blue. 

Yellow. 

72.8 

Neither  so  rich 

colours  as  the  last. 

Purple  (rather  pale.) 

Pale  yellow. 

Blue. 

Yellow. 

71.7 

Coming  up  to  good  colours, 

the   yellow    a  better  colour 

than  a  gilt  picture-frame. 

Fine  pink. 

Fine  green. 

Yellow,  but  has  got  a  good 

Blue,  but  has  a  good  deal  of 

69.7 

deal  of  blue  in  it. 

yellow  in  it. 

Fine  yellow. 

Purple. 

Good  yellow. 

Good  blue. 

68.2 

Better  colours  than 

any  yet  seen. 

Yellowish  green. 

Fine  crimson. 

Yellow,  but  has  a  good  deal 

Blue,  but  has  a  good  deal  of 

67.0 

of  blue. 

yellow. 

Good  blue,  verging  to  in- 

Yellow, verging  to  orange. 

Blue. 

Yellow. 

65.5 

digo. 
Red,  or  very  ruddy  pink. 

Very    pale    greenish   blue, 

Both  gay  colours,  particularly 
Yellow. 

the  yellow  to  the  riglit. 
Blue. 

63.8 

almost  white. 

Rich  yellow. 

Full  blue. 

Fine  bright  yellow. 

Pretty  good  blue. 

627 

White. 

Fie.-y  orange. 

Has  very  little  colour. 

Yellow,  but  a  different  vel- 

61.2 

low,  it  is  a  blood-looking 

Dark  purple. 

While. 

A  dim  blue,  wants  light. 

yellow. 
White,  with  a  dash  of  yel- 

59.5 

low  and  blue. 

Dull  orange  red. 

White. 

Yellow 

White,  with  blue  and  yel- 

59.0 

low  in  it. 

White. 

Dull  dirty  olive. 

White. 

Dark. 

57.1 

Very  dark  purple. 

White. 

Dark. 

White. 

55.0 

Part  II. 


Instead  of  presenting  the  colours  for  his  judgment,  he  was  now  desired  to  arrange  the  apparatus  so  as  to       508. 
make  the  strongest  possible  succession  of  contrasts  of  colour  in  the  two  circles.     The  results  were  ;  s  follow  : 


Colours  according  to  the  judgment  of 
an  ordinary  eye. 

Colours  as  named  by  the  individual 
in  question. 

Inclination 
of  the 
plate  of 
mica  to  eye. 

Circle  to  the  left. 

Circle  to  the  right. 

Circle  to  the  left. 

Circle  to  the  right. 

Pale  ruddy  pink. 
Blue  green. 
Yellow. 
White. 
Pale  brick-red. 
Indigo. 
Yellow. 

Blue  green. 
Pale  ruddy  pink. 
Blue. 
Fiery  orange. 
White. 
Pale  yellow. 
Indigo. 

Yellow. 
Blue. 
Yellow. 
Blue. 
Yellow. 
Blue. 
Yellow. 

Blue. 
Yellow. 
Blue. 
Yellow. 
Blue. 
Yellow. 
Blue. 

59.1° 
65.3 
63.1 
61.1 
58.5 
542 
52.1 

It  appears  by  this,  that  the  eyes  of  the  individual  in  question  are  only  capable  of  fully  appreciating  blue  and 
yellow  tints,  and  that  these  names  uniformly  correspond,  in  his  nomenclature,  to  the  more  and  less  refrangible 
rays,  generally  ;  all  which  belong  to  the  former,  indifferently,  exciting  a  sense  of  "  blueness,"  and  to  the  tatter 
of  "  yellowness."  Mention  has  been  made  of  individuals  seeing  well  in  other  respects,  but  devoid  altogether 
of  the  sense  of  colour,  distinguishing  different  tints  only  as  brighter  or  darker  one  than  another;  but  the  case 
is,  probably,  one  of  extremely  rare  occurrence. 

Mayer,  in  an  Essay  De  Affinitate  Colorum,  (Opera  inedita,  1775,)  regards  all  colours  as  arising  from  three 
primary  ones,  red,  yellow,  and  blue  ;  regarding  white  as  a  neutral  mixture  of  rays  of  all  colours,  and  black  as  a 
mere  negation  of  light.  According  to  this  idea,  were  we  acquainted  with  any  mode  of  mixing*  colours  in 
simple  numerical  ratios,  a  scale  might  be  formed  to  which  any  proposed  colour  might  be  at  once  referred.  He 
proposes  to  establish  such  a  scale  in  which  the  degrees  of  intensity  of  each  simple  colour  shall  be  represented 
by  the  natural  numbers  1,  2,  3.  ...  12;  1  denoting  the  lowest  degree  of  it  capable  of  sensibly  affecting  a  tint, 
and  12  the  full  intensity  of  which  the  colour  is  capable,  or  the  total  amount  of  it  existing  in  white  light.  Thus 
r14  denotes  a  full  red  of  the  brightest  and  purest  tint,  y1'1  the  brightest  yellow,  and  612  the  brightest  blue.  To 
represent  mixed  tints,  he  combines  the  symbols  of  the  separate  ingredients.  Thus  r14  y4,  or,  more  convenientlyf 
12  r  -f  4  y,  represents  a  red  verging  strongly  to  orange,  such  as  that  of  a  coal  fire. 

The  scale  proposed  is  convenient  and  complete,  so  far  as  regards  what  he  calls  perfect  colours,  which  arise 
from  white  light  by  the  subtraction  of  one  or  more  proportions  of  its  elementary  rays  ;  but  a  very  slight  moditi- 

3  I  2 


509. 

Mayer's 
hypothesis 


colour* 


Codification 
Of  Mayer's 
scale. 


436 


LIGHT. 


Lifht. 


511. 

Whites, 
greys,  and 
neutral 
tints. 


512. 

Reds,  yel- 
lows, and 

olues. 


513. 


514. 

Browns. 


515. 

Purplos. 


516. 

Greens. 


517. 

The  same 
colour  pro 
duced  by 
different 
prismatic 
combina- 
tion!. 


cation  of  his  system  will  render  it  equally  applicable  to  all,  and  it  may  be  presented  as  follows.     Suppose  we     P»rt  II. 
fix  on  100  as  a  standard  intensity  of  each  primary  colour;  or  the  number  of  rays  of  that  colour  (all  supposed  v-— v~^" 
equally  effective)   which  falling  on   a  sheet  of  white  paper,  or  other  surface  perfectly  neutral,  (i.  e.  equally 
disposed  to  reflect  all   rays)  shall  produce  a  full  tint  of  that  particular  kind,  and  let  us  denote  by  such  an 
expression  as  x  R  +  y  Y  +  z  B,  the  tint  produced  by  the  incidence  of  x  such  rays  of  primary  red,  y  such  ruys 
of  yellow,  and  z  such  rays  of  blue  on  the  same  surface  together.     It  is  obvious  then,  that  the  different  numerical 
values  assigned  to  x,  y,  2,  from  1  to   100,  will  give  different  symbols  of  tints,  whose  number  will  be  100   x 
100  x   100  =  1000000,  and  therefore  quite  sufficient  in  point  of  extent  to  embrace  all  the  variety  of  colours 
the  eye  can  distinguish.     The  number  of  tints  recognised  as  distinct  by  the  Roman  artists  in  Mosaic  is  said 
to  exceed  30,000  ;  but  if  we  suppose  ten  times  this  amount  to  occur  in  nature  (and   it  is  obvious  that  these 
must  be  greatly  more  numerous  than  the  purposes  of  the  painter  admit)   we  are  still  much  within  the  limits  of 
our  scale.     It  only  remains  to  examine  how  far  the  tints  themselves  are  expressible  by  the  members  of  the  scale 
proposed. 

And  first,  then,  of  whites,  greys,  and  neutral  tints.  The  most  perfectly  neutral  tints,  which  are,  in  fact,  only 
greater  and  less  intensities  of  whiteness,  are  those  we  observe  in  the  clouds  in  an  ordinary  cloudy  day,  with 
occasional  gleams  of  sunshine.  From  the  most  sombre  shadows  to  the  snowy  whiteness  of  those  cumulus- 
shaped  clouds  on  which  the  sun  immediately  shines,  we  have  nothing  but  a  series  of  whites,  or  greys,  repre- 
sented by  such  combinations  as  R  +  Y  +  B,  2  R+2  Y  +  2  B,  &c.  ;  or  n  (R  +  Y  +B)  which,  for  brevity,  we  may 
represent  by  n  W.  To  be  satisfied  of  this  we  need  only  look  through  a  tube  blackened  on  the  inside  to  prevent 
surrounding  objects  influencing  our  judgments  ;  and  any  small  portion  thus  insulated  of  the  darkest  clouds 
will  appear  to  differ  in  no  respect  from  a  portion  similarly  insulated  of  a  sheet  of  white  paper  more  or  less 
shaded. 

The  various  intensities  of  pure  reds,  yellows,  and  blues  are  represented  by  n  R,  n  Y,  and  n  B  respectively. 
They  are  rare  in  nature ;  but  blood,  fresh  gilding,  or  gamboge  moistened,  and  ultramarine  may  be  cited  as 
examples  of  them.  Scarlets  and  vivid  reds,  such  as  vermilion  and  minium,  are  not  free  from  a  mixture  of 
yellow,  and  even  of  blue ;  for  all  the  primary  colours  are  greatly  increased  in  splendour  by  a  certain  mixture 
of  white,  and  whenever  any  primary  colour  is  peculiarly  glaring  and  vivid,  we  may  be  sure  that  it  is  in  some 
degree  diluted  with  white.  The  blue  of  the  sky  is  white,  with  a  very  moderate  addition  of  blue. 

The  mixture  of  red  and  yellow  produces  all  the  shades  of  scarlet,  orange,  and  the  deeper  browns,  when  of 
feeble  intensity.  When  diluted  with  white,  we  have  lemon  colour,  straw  colour,  clay  colour,  and  all  the  brighter 
browns  ;  the  last-mentioned  tints  growing  duskier  and  dingier  as  the  coefficients  are  smaller. 

The  browns,  however,  are  essentially  sombre  tints,  and  produce  their  effects  chiefly  by  contrast  with  other 
brighter  hues  in  their  neighbourhood.  To  produce  a  brown,  the  painter  mixes  black  and  yellow,  or  black  and 
red,  (that  is,  such  impure  reds  as  the  generality  of  red  pigments,)  or  all  three;  his  object  is  to  stifle  light, 
and  leave  only  a  residuum  of  colour.  There  it  a  brown  glass  very  common  in  modern  ornamental  windows. 
If  examined  with  a  prism,  it  is  found  to  transmit  the  red,  orange,  and  yellow  rays  abundantly,  little  green,  and 
no  pure  blue.  The  small  quantity  of  blue,  then,  that  its  tint  does  involve,  must  be  that  which  enters  as  a 
component  part  of  its  green,  (in  this  view  of  the  composition  of  colours,)  and  its  characteristic  symbol  may 
thus  be,  perhaps,  of  some  such  form  as  10  R  -f-  9  Y  +  1  B  ;  that  is  to  say,  (9  R  +  8  Y)  +  I  (R  +  Y  +  B),  or  an 
orange  of  the  character  9  R  +  8  Y  diluted  with  one  ray  of  white.  It  must  be  confessed,  however,  that  the 
composition  of  brown  tints  is  the  least  satisfactory  of  all  the  applications  of  Mayer's  doctrine.  He  himself  has 
passed  it  unnoticed. 

Combinations  of  red  and  blue,  and  their  dilutions  with  white,  form  all  the  varieties  of  crimson,  purple,  violet, 
rose  colour,  pink,  &c.  The  richer  purples  are  entirely  free  from  yellow.  The  prismatic  violet,  when  compared 
with  the  indigo,  produces  a  sensible  impression  of  redness,  and  must  therefore  be  regarded  on  this  hypothesis 
as  consisting  of  a  mixture  of  blue  and  red  rays. 

Blue  and  yellow,  combined,  produce  green.  The  green  thus  arising  is  vivid  and  rich  ;  and,  when  proper 
proportions  of  the  elementary  colours  are  used,  no  way  to  be  distinguished  from  the  prismatic  green.  Nothing 
can  be  more  striking,  and  even  surprising,  than  the  effect  of  mixing  together  a  blue  and  a  yellow  powder,  or 
of  covering  a  paper  with  blue  and  yellow  lines,  drawn  close  together,  and  alternating  with  each  other.  The 
elementary  tints  totally  disappear,  and  cannot  even  be  recalled  by  the  imagination.  One  of  the  most  marked 
facts  in  favour  of  the  idea  of  the  existence  of  three  primary  colours,  and  of  the  possibility  of  an  analysis  of 
white  light  nistinct  from  that  afforded  by  the  prism,  is  to  see  the  prismatic  green  thus  completely  imitated  by 
a  mixture  of  adjacent  rays  totally  distinct  from  it,  both  in  refrangibility  and  colour. 

The  hypothesis  of  three  primary  colours,  of  which,  in  different  proportions,  all  the  colours  of  the  spectrum 
are  composed,  affords  an  easy  explanation  of  a  phenomenon  observed  by  Newton,  viz.  that  tints  no  way 
distinguishable  from  each  other  may  be  compounded  by  very  different  mixtures  of  the  seven  colours  into  which 
he  divided  it.  Thus  we  may  regard  white  light,  indifferently,  as  composed  of 

b  rays  of  pure  red  =  R' 


R  =  a  +  b  +  c  rays  of  pure  red  ~i 

Y  =  d  +  e-J-/-r-g  rays  of  pure  yellow  >  or  of 

B  =  ft  +  i  +  k  +  I  rays  of  pure  blue    J 


c  +  d  rays  of  orange  (c  red  +  d  yellow)  =  O 

e  rays  of  pure  yellow  =  Y' 

/  +  h  rays  of  green  (/yellow  +  h  blue)  =  G' 

g  +  i  rays  of  prismatic  blue  (g  yellow  +  i  blue)  =  B 

k  rays  of  indigo,  or  pure  blue  =  I' 

;  +  a  rays  of  violet  (I  blue  +  a  red)  =  V 


LIGHT.  437 

Light,     and  any  tint  capable  of  being1  represented  by  x  .  R  +  y  .  Y  +  z  B,  may  be  represented  equally  well  by  Part  II- 

m  .  R'  +  n  .  0'  +  p  .  Y'  +  q  ,  G'  +  r  .  B'  +  s  .  1'  +  t  .  V, 
provided  we  assume  m,  n,  p,  &c.,  such  as  to  satisfy  the  equations 

mb  +  n  c  •+  ta  —  x;        n  d  +  p  e  +  qf  +  rg  =  y ;        g  h  4-  r  i  +  s  k  +  tl  —  2. 

From  what  has  been  said  we  shall  now  proceed  to  show,  that,  without  departing  from  Mayer's  doctrine,  any       518. 
other  three  prismatic  rays  may  still  be   equally  assumed  as  fundamental  colours,  and  all  the  rest  compounded  f>r'Y°ulfi 
from  them,  provided  we  attend  only  to  the  predominant  tint  resulting,  and  disregard  its  dilution  with  white.  Jjf 'JJ^*18 
For  instance,  Dr.  Young  has  assumed  red,  green,  and  violet  as  his  fundamental  colours;    and  states,  as  an  otherprima- 
experimental  fact  in  support  of  this  doctrine,  that  the  perfect  sensations  of  yellow  and  blue  may  be  produced,  ry  colours, 
the  former  by  a  mixture  of  red  and  green,  and  the  latter  by  green  and  violet.  (Lectures  on  Natural  Philosophy, 
p.  439.)     Now,  if  we  mix  together  yellow  and  white  in  the  proportion  of  m  yellow  +  n  white,  the  compound 
will  produce  a  perfect  sensation  of  yellow,  unless  m  be  small   compared  to  n  ;    but,  assuming  white  to  be 
composed  as  above,  this  compound  is  equivalent  to 

n  R  red  +  (m  +  n)  Y  yellow  +  n  B  blue. 

On  the  other  hand,  if  we  mix  together  P  such  red  rays  (each  of  the  intensity  6)  and  Q  such  green  rays  (each 
consisting  of  yellow,  of  the  intensity/  and  blue  of  the  intensity  A)  as  are  supposed  in  the  foregoing  article  to 
exist  in  the  spectrum,  we  have  a  compound  of 

P  .  b  red  +  Q  .  /yellow  +  Q  .  A  blue, 
and  these  will  be  identical  with  the  former,  if  we  take 

nR=P6;        («i  +  «)Y=Q/;        nB  =  QA. 
Eliminating  Q  from  the  two  last  of  these,  we  get 

^L     L    JL 

n       '    h    '     Y 

for  the  relation  between  M  and  N.  Now  the  only  conditions  to  be  satisfied  are  that  M  shall  be  positive,  and 
not  much  less  than  N  ;  and  it  is  evident  that  these  conditions  may  be  fulfilled  an  infinite  number  of  ways  by  a 
proper  assumption  of  the  ratio  of  /to  A.  In  the  same  manner,  if  we  suppose  a  mixture  of  M  rays  primary 
blue  =  B  with  N  rays  of  white  (=  R  +  Y  +  B)  to  be  equivalent  to  P  rays  of  prismatic  green  mixed  with  Q 
of  violet,  we  get  the  equation 

m         ^        R  h         Y 

n    ''"   a    '  TT     '    /         B 

Suppose,  for  example,,  we  regard  white  light  as  consisting  of  20  rays  of  primary  red,  30  of  yellow,  and  50      519. 
of  blue,  and  the  several  prismatic  rays  to  consist  as  follows:  Numeric»l 

'lluitration. 
Red         8  rays  primary  red  —  A. 

Orange   7   red  +  7  primary  yellow  =  c  +  d. 

Yellow    8 yellow  =  e. 

Green    10 yellow  +  10  primary  blue  =  /+  A. 

Blue        6 yellow  +  12  primary  blue  =  g  +  i. 

Indigo  12 blue  =  k. 

Violet    16   blue  +  5  primary  red  =  I  +  a. 

Then  will  the  union  of  15  rays  of  such  red  with  30  of  such  green,  produce  a  compound  ray  containing 
15  x  8  =  120  of  primary  red,  30 x  10  =  300  of  primary  yellow,  and  30  x  10  =  300  of  primary  blue;  which 
are  the  same  as  exist  in  a  yellow,  consisting  of  6  rays  of  white  combined  with  4  of  primary  yellow,  'in  like 
manner,  if  75  such  green  rays  be  combined  with  100  such  violet,  the  result  will  be  100  x  5  =  500  rays  of 
primary  red,  +  75  x  10  =  750  of  primary  yellow,  +  75  x  10  +  100  x  16  =  2350  of  primary  blue,  which 
together  compose  a  tint  identical  with  that  which  would  result  from  the  union  of  25  rays  of  white  with  22  of 
primary  blue ;  that  is  to  say,  a  fine  lively  blue.  The  numbers  assumed  above,  it  must  be  understood,  are 
merely  taken  for  the  sake  of  illustration,  and  are  no  way  intended  to  represent  the  true  ratios  of  the  differently 
coloured  rays  in  the  spectrum. 

The  analogy  of  the  fixed  lines  in  the  solar  spectrum  might  lead  us  to  look  for  similar  phenomena  in  other 
sources  of  light.     Accordingly,  Fraunhofer  has  found,  that  each  fixed  star  has  its  own  particular  system  of  dark      520- 
and  bright  spaces  in  its  spectrum ;  but  the  most  curious  phenomena  are  those  presented  by  coloured  flames  Pfhen°mena 
which  produce  spectra  (when  transmitted  through  a  colourless  prism)  hardly  less  capricious  than  those  afforded          ' 
oy  solar  light  transmitted  through  coloured  glasses.    Dr.  Brewster,  Mr.  Talbot,  and  others,  have  examined  these 


438 


LIGHT. 


Light. 


521. 

Flames  of 
combusti- 
bles burning 
feebly. 


522. 

Burniug 
strongly. 


523. 

Flames 
coloured  by 
saline 
bodies. 


524. 

Die  colour 
depends 
shiefly  on 
the  base. 


phenomena  with  attention;   but  the  subject  is  not  exhausted,  and  promises  a  wide  field  of  curious  research.      Part  II. 
The  following:  facts  may  be  easily  verified :  v>~v"1*p' 

1.  Most  combustible  bodies  consisting  of  hydrogen  and  carbon,  as  tallow,  oil,  paper,  alcohol,  &c.  when 
first  lighted  and  in  a  state  of  feeble  and  imperfect  combustion,  give  blue  flames.     These,  when  examined  by 
the  prism,  by  letting  them  shine  through  very  narrow  slits  parallel  to  its  edge,  as  described  in  Art.  487,  all  give 
interrupted  spectra,  consisting,  for  the  most  part,  of  narrow  lines  of  very  definite  refrangibility,  either  separated 
by  broad  spaces  entirely  dark,  or  much  more  obscure  than  the  rest.     The  more  prominent  rays  are,  a  very  narrow 
definite  yellow,  a  yellowish  green,  a  vivid  emerald  green,  a  faint  blue,  and  a  strong  and  copious  violet. 

2.  In  certain  cases  when   the  combustion  is  violent,  as  in  the  case  of  an  oil  lamp  urged  by  a  blow-pipe, 
(according  to  Fraunhofer,)  or  in  the  upper  part  of  the  flame  of  a  spirit  lamp,  or  when  sulphur  is  thrown  into 
a  white-hot  crucible,  a  very  large  quantity  of  a  definite  and  purely  homogeneous  yellow  light  is  produced ;  and 
in  the  latter  case  forms  nearly  the  whole  of  the  light.     Dr.  Brewster  has  also  found  the  same  yellow  light  to  be 
produced  when  spirit  of  wine,  diluted  with  water  and  heated,  is  set  on  fire ;  and  has  proposed  this  as  a  means 
of  obtaining  a  supply  of  homogeneous  yellow  light  for  optical  experiments. 

3.  Most  saline  bodies  have  the  power  of  imparting  a  peculiar  colour  to  flames  in  which  they  are  present, 
either  in  a  solid  or  vaporous  state.     This  may  be  shown  in  a  manner  at  once  the  most  familiar  and  most  effi- 
cacious, by  tHe  following  simple  process :  Take  a  piece  of  packthread,  or  a  cotton  thread,  which  (to  free  it  from 
saline  particles  should  have  been  boiled  in  clean  water,)  and  having  wetted  it,  take  up  on  it  a  little  of  the  salt 
to  be  examined  in  fine  powder,  or  in  solution.     Then  dip  the  wetted  end  of  it  into  the  cup  of  a  burning  wax 
candle,  and  apply  it  to  the  exterior  of  the  flame,  not  quite  in  contact  with  the  luminous  part,  but  so  as  to  be 
immersed  in  the  cone  of  invisible  but  intensely-heated  air  which  envelopes  it.     Immediately  an  irregular  sput- 
tering combustion  of  the   wax  on  the  thread  will  take  place,  and  the  invisible  cone  of  heat  will   be  rendered 
luminous,  with  that  particular  coloured  light  which  characterises  the  saline  matter  employed. 

Thus  it  will  be  found  that,  in  general, 

Salts  of  soda  give  a  copious  and  purely  homogeneous  yellow. 

Salts  of  potash  give  a  beautiful  pale  violet. 

Salts  of  lime  give  a  brick  red,  in  whose  spectrum  a  yellow  and  a  bright  green  line  are  seen. 

Salts  of  strontia  give  a  magnificent  crimson.     If  analyzed  by  the  prism  two  definite  yellows  are  seen,  one 

of  which  trerges  strongly  to  orange, 
Salts  of  magnesia  give  no  colour. 

Salts  of  lithia  give  a  red,  (on  the  authority  of  Dr.  Turner's  experiments  with  the  blow-pipe.) 
Salts   of  baryta  give  a  fine  pale  apple-green.     This  contrast  between  the  flames  of  baryta  and   strontia  is 

extremely  remarkable. 

Salts  of  copper  give  a  superb  green,  or  blue  green. 
Sa't  of  iron  (protoxide)  gave  white,  where  the  sulphate  was  used. 

Of  all  salts,  the  muriates  succeed  best,  from  their  volatility.  The  same  colours  are  exhibited  also  when  any  of 
the  salts  in  question  are  put  (in  powder)  into  the  wick  of  a  spirit  lamp.  If  common  salt  be  used,  Mr.  Talbot 
has  shown  that  the  light  of  the  flame  is  an  absolutely  homogeneous  yellow ;  and,  being  at  the  same  time  very 
copious,  this  property  affords  an  invaluable  resource  in  optical  experiments,  from  the  great  ease  with  which  it 
is  obtained,  and  its  identity  at  all  times.  The  colours  thus  communicated  by  the  different  bases  to  flame,  afford 
in  many  cases  a  ready  and  neat  way  of  detecting  extremely  minute  quantities  of  them  ;  but  this  rather  belongs 
to  Chemistry  than  to  our  present  subject.  The  pure  earths,  when  violently  heated,  as  has  recently  been  prac- 
tised by  Lieutenant  Drummond,  by  directing  on  small  spheres  of  them  the  flames  of  several  spirit  lamps  urged 
by  oxygen  gas,  yield  from  their  surfaces  lights  of  extraordinary  splendour,  which,  when  examined  by  prismatic 
analysis,  are  found  to  possess  the  peculiar  definite  rays  in  excess,  which  characterise  the  tints  of  flames  coloured 
by  them  ;  so  that  there  can  be  no  doubt  that  these  tints  arise  from  the  molecules  of  the  colouring  matter  reduced 
to  vapour,  and  held  in  a  state  of  violent  ignition. 


LIGHT  439 


PART  III. 

OF  THE  THEORIES  OF  LIGHT. 

Light.  AMONG  the  theories  which  philosophers  have  imagined  to  account  for  the  phenomena  of  light,  two  principally  pari  m 
— •v-^*'  have  commanded  attention ;  the  one  conceived  by  Newton,  and  called  from  his  illustrious  name,  in  which  light  -_i- v  - 
is  conceived  to  consist  of  excessively  minute  molecules  of  matter  projected  from  luminous  bodies  with  the  525. 
immense  velocity  due  to  light,  and  acted  on  by  attractive  and  repulsive  forces  residing  in  the  bodies  on  which 
they  impinge,  which  turn  them  aside  from  their  rectilinear  course,  and  reflect  and  refract  them  according  to 
the  laws  observed.  The  other  hypothesis  is  that  of  Huygens,  and  also  called  after  his  name ;  which  supposes 
light  to  consist,  like  sound,  in  undulations,  or  pulses,  propagated  through  an  elastic  medium.  This  medium  is 
conceived  to  be  of  extreme  elasticity  and  tenuity  ;  such,  indeed,  that  though  rilling  all  space,  it  shall  offer  no 
appreciable  resistance  to  the  motions  of  the  planets,  comets,  &c.  capable  of  disturbing  them  in  their  orbits.  It 
is,  moreover,  imagined  to  penetrate  all  bodies  ;  but  in  their  interior  to  exist  in  a  different  state  of  density  and 
elasticity  from  those  which  belong  to  it  in  a  disengaged  state,  and  hence  the  refraction  and  reflexion  of  light. 
These  are  the  only  mechanical  theories  which  have  been  advanced.  Others,  indeed,  have  not  been  wanting ; 
such  as  Professor  Oersted's,  who,  in  one  of  his  works,  considers  light  as  a  succession  of  electric  sparks,  or  a 
series  of  decompositions  and  recompositions  of  an  electric  fluid  filling  all  space  in  a  neutral  or  balanced  state, 
&c.  &c.  In  this  part,  however,  we  propose  only  to  give  an  account  of  the  Newtonian  and  Huygenian  theories, 
so  far  as  they  apply  to  the  phenomena  already  described  ;  and  thus  prepare  ourselves  for  the  remaining  more 
complex  branches  of  the  History  of  the  Properties  of  Light,  which  can  hardly  be  understood,  or  even  described, 
without  a  reference  to  some  theoretical  views. 


§  I.    Of  the  Newtonian  or  Corpuscular  Theory  of  Light. 

Postulata.    1.  That  light  consists  of  particles  of  matter  possessed  of  inertia    and  endowed  with  attrac-       535 
live  and  repulsive  forces,  and  projected  or  emitted  from  all  luminous  bodies  with    nearly  the    same    velocity, 
about  200,000  miles  per  second. 

2.  That  these  particles  differ  from  each  other  in  the  intensity  of  the  attractive  and  repulsive  forces  which 
reside  in  them,  and  in  their  relations  to  the  other  bodies  of  the  material  world,  and  also  in  their  actual  masses, 
or  inertia. 

3.  That  these   particles,  impinging  on  the  retina,   stimulate  it  and  excite  vision.      The    particles   whose 
inertia  is  greatest  producing  the  sensation  of  red,  those  of  least  inertia  of  violet,  and  those  in  which  it  is  inter- 
mediate the  intermediate  colours. 

4.  That  the  molecules  of  material  bodies,  and  those  of  light,  exert  a  mutual  action  on  each  other,  which 
consists  in  attraction  and  repulsion,  according  to  some  law  or  function  of  the  distance  between  them  ;  that  this 
law  is  such  as  to  admit,  perhaps,  of  several  alternations,  or  changes  from   repulsive  to  attractive  force ;  but 
that  when  the  distance  is  below  a  certain  very  small  limit,  it  is  always  attractive  up  to  actual  contact ;  and  that 
beyond  this  limit  resides  at  least  one  sphere  of  repulsion.     This   repulsive  force  is  that  which  causes   the 
reflexion  of  light  at  the  external  surfaces  of  dense  media ;   and  the  interior  attraction  that  which  produces  the 
refraction  and  interior  reflexion  of  light. 

5.  That  these  forces  have  different  absolute  values,  or  intensities,  not  only  for  all  different  material  bodies, 
but  for  every  different  species  of  the  luminous  molecules,  being  of  a  nature  analogous  to  chemical  affinities,  or 
electric  attractions,  and  that  hence  arises  the  different  refrangibility  of  the  rays  of  light. 

6.  That  the  motion  of  a  particle  of  light  under  the  influence  of  these  forces  and  its  own  velocity  is  regu- 
lated by  the  same  mechanical  laws  which  govern  the  motions  of  ordinary  matter,  and  that  therefore  each  particle 
describes  a  trajectory  capable  of  strict  calculation  so  soon  as  the  forces  which  act  on  it  are  assigned. 

7.  That  the  distance  between  the  molecules  of  material  bodies  is  exceedingly  small  in  comparison  with  the 
extent  of  their  spheres  of  attraction  and  repulsion  on  the  particles  of  light.     And 

8.  That  the  forces  which  produce  the  reflexion  and  refraction  of  light  are,  nevertheless,  absolutely  insensible 
at  all  measurable  or  appreciable  distances  from  the  molecules  which  exert  them. 

9.  That  every  luminous  molecule,  during  the  whole  of  its   progress  through  space,  is  continually  passing 
through  certain   periodically  recurring  states,  called  by  Newton  fits  of  easy  reflexion  and  easy  transmission,  in 
virtue  of  which  (from  whatever  cause  arising,  whether  from  a  rotation  of  the  molecules  on  their  axes,  and  the 
consequent  alternate  presentation  of  attractive  and  repulsive  poles,  or  from  any  other  conceivable  cause)  they 
are  more  disposed,  when  in  the  former  states  01  phases  of  their  periods,  to  obey  the  influence  of  the  repulsive 
or  reflective  forces  of  the  molecules  of  a  medium  ;  and  when  in  the  latter,  of  the  attractive.     This  curious  and 
delicate  part  of  the  Newtonian  doctrine  will  be  developed  more  at  large  hereafter. 


440  LIGHT. 

Ijpit  It  is  the  7th  and  8th  of  these  assumptions  only  which  render  the  course  pursued  by  a  luminous  molecule,    ^art 

^—  —  v""""'  under  the  influence  of  the  reflective  or  refractive  forces,  capable  of  being  reduced  to  mathematical  calculation  ;  ^~ 

527.       for  it  follows  immediately  from  the  8th,  that,  up  to  the  very  moment  when  such  a  molecule  arrives  in  physical 

contact  with  the  surface  of  any  medium,  it  is  acted  on  by  no  sensible  force,  and  therefore  not  sensibly  deviated 

from  its  rectilinear  path  ;  and,   on  the  other  hand,  as  soon  as  it  has  penetrated  to  any  sensible  depth  within  the 

surface,  or  among  the  molecules,  by  reason  of  the  7th  of  the  above  postulates,  it  must  be  equally  attracted  and 

repelled  by  them  in  all  directions,  and  therefore  will  continue  to  move  in  a  right  line,  as  if  under  the  influence 

of  no  force.     It  is  only,  therefore,  within  that  insensible  distance  on  either  side  the  surface,  which  is  measured 

by  the  diameter  of  the  sphere  of  action  of  each  molecule,  that  the  whole  flexure  of  the  ray  takes  place.     Its 

trajectory  then  may  be  regarded  as  a  kind  of  hyperbolic  curve,  in  which  the  right  lines  described  by  it,  previous 

and  subsequent  to  its  arrival  at  the  surface,  are  the  infinite  branches,  and  are  confounded  with  the  asymptotes, 

and  the  curvilinear  portion  is  concentered  as  it  were  in  a  physical  point.     Now,  in  explaining  the  phenomena 

of  reflexion  and  refraction,  it  is  not  the  nature  of  this  curve  that  we   are  called  on  to  investigate.     This  will 

depend  on  the  laws  of  corpuscular  action,  and  must  necessarily  be  of  great  complexity.     All  we  have  to  inquire, 

is  the  direction  the  ray  will  ultimately  take  after  incidence,  and  the  final  change,  if  any,  in  its  velocity. 

b28.  Let  us,  then,  consider  the  motion  of  a  molecule  urged  to  or  from  the  surface  of  a  medium   by  the  united 

Motion  of  a  attractions  or  repulsions  of  all  its  particles  acting  according  to  any  conceivable  mathematical  law.     And,  first, 

luminous       jj  js  evident,  that  supposing  the  surface  mathematically  smooth,  and  the  number  of  attractive   or   repulsive 

'der"the     particles    of  which   it   consists,   infinite,  their   total  resultant    force  on  the  luminous   molecule   will  act  in  a 

influence  of  direction  perpendicular  to  the  surface  ;  and  will  be  insensible  at  all  sensible  distances  from  the  surface,  provided 

»ny  forces,    the  elementary  forces  of  each  molecule  decrease  with  sufficiently  great  rapidity  as  the  distances  increase.     This 

condition  being  supposed,  let  x  and  y  be  the  coordinates  of  the  molecule  at  any  assigned  instant  ;  the  plane  of 

the  x  and  y  being  supposed  to  coincide  with  that  of  its  trajectory,  out  of  which  plane  there  is  evidently  no  force 

to  turn  it,  and  which  must  of  course  be  perpendicular  to  the  surface  of  the  medium  in  which  x  is  supposed 

to  lie  :    y  then  will  be  the  perpendicular  distance  of  the  luminous  molecule  from   this   surface,  and  Y  (some 

function  of  y  decreasing  with  extreme  rapidity)  will  represent  the  force  urging  it  inwards,  or  towards  the  surface 

when  the  molecule  is  without,  from  when  within  the  medium.     Therefore,  by  the  principles  of  Dynamics,  sup. 

posing  d  t  to  denote  the  element  of  the  time,  we  shall  have  for  the  equations  of  the  motion 


and  hence,  multiplying  the  first  by  dx,  the  second  by  dy,  adding  and  integrating,  we  get 


dx*  +  d  y* 


/»,. 

2   /  Y  d  y  = 


,. 

+  2   /  Y  d  y  =  constant. 


Now,  t)  being  the  velocity  of  the  molecule,  we  have  u5  =  — -,  and  therefore  this  equation  becomes 


-2/1 


c>  =  constant  —  2    /  Y  d  y. 

It  is,  however,  only  with  the  terminal  velocity,  or  that  attained  by  the  light  after  undergoing  the  total  action  of 
the  medium,  that  we  are  concerned,  and  therefore  if  we  put  V  for  its  primitive,  or  initial,  and  V  for  its  terminal 
velocity,  we  shall  have,  by  extending  the  integral  from  the  value  of  y  at  the  commencement  of  the  ray's  motion 
(y0)  to  its  value  at  the  end  (y,), 

V'«-  V«  =  -  2/Ydy.  (6) 

Since  ya  and  y/  are  supposed  infinite,  and  since  the  function  Y  decreases  by  hypothesis  with  such  rapidity  as  to 
become  absolutely  insensible  for  all  sensible  values  of  y,  it  is  clear  that  we  may  take  y?  =  +  cc  for  the  first 
limit  of  the  integral  in  all  cases.  With  regard  to  the  other,  we  have  now  to  distinguish  two  principal 
cases : 

529.  The  first  is  that  of  reflexion,  where  the  ray,  no  matter  whether  before  its  arrival  at  the  surface,  or  at  reaching 

Cue  of  re.  it,  or  even  after  passing  some  small  distance  into  the  medium,  is  turned  back  by  the  prevalence  of  the  repulsive 

flexion.         force,  and  pursues  the  whole  of  its  course  afterwards  without  the  medium.     Now  in  this  case  if  we  resolve  the 

integral  fYdy  into  its  elements,  these,  in  the  approach  of  the  molecule  to  the  surface,  may  be  represented  as 

follows, 

&c.  +  Y'  x  -  d  y  +  Y"  x  -  d  y  +  Y"  x  -  dy  +&c 

But  in  the  recess  of  the  molecule,  the  values  of  y  increase  again  by  the  same  steps  as  they  before  diminished 
and  become  identical  with  the  former  ones;  and  Y',  Y",  &c.,  the  values  of  Y  corresponding  to  the  successive 
values  of  y,  remain  therefore  the  same,  both  in  size  and  magnitude ;  the  corresponding  elements  of  the  integral 
generated  during  the  recess  of  the  molecule  will  be  then 

&c.  +  Y'  x  -f  d  y  +  Y"  X  +  d  y  +  Y'"  X  +  d  y  +  &c. 


LIGHT.  441 

1;«ht-       So  tnat,  co .nbiiiing  boili,  the  latter  exactly  destroy  the  former,  and  givey  Y  dy  —  0  when  extended  from  one  end 
--~\— ~'  to  the  other  of  the  trajectory.     Thus  we  have,  in  the  case  of  reflexion, 

V'i  _  V*  =  0,         or  V  =  V. 

The  second  case  is  that  in  which  the  whole  course  of  the  ray  after  incidence  lies  within  the  medium,  or  the  case        530. 
of  refraction.     Here  the  values  of  y  before  incidence  are  all  positive,  and  after,  all  negative;  and,  moreover,  the  Case  of 
change  of  sign  in  dy  which  happened  in  the  case  of  reflexion,  does  not  here  take  place.     Hence  J"Y  dy  must  ^fraction. 
be  extended  from  -j-  GO  to  —  co ,  and  its  value  will  not  vanish,  but  (on   account  of  the  rapid  decrease  of  the 
function  Y)  will  have  some  finite  value.     Now  this  can  only  he  dependent  on  the  arbitrary  quantities  which 
enter  into  the  composition  of  Y ;  in  other  words,  on  the  nature  of  the  medium  and  the  ray,  and  not  at  all  on  the 
constants  which  determine  the  direction  of  the  ray  with  respect  to  the  surface,  (as  its  inclination  or  the  position 
of  the  plane  of  incidence.)      Hence  we  may  suppose  y  Ydi/  =  —  ^k  V*,  where  ft  is  a  constant  independent  of 
the  direction  of  the  ray,  and  determined  only  by  its  nature  and  that  of  the  medium,  and  we  shall  have 


(c) 


putting  v'l  -J-  ft  =  fi. 

Hence  we  see  that  both  in  refraction  and  reflexion,  on  this  hypothesis,  the  velocity  of  the  ray  after  deviation       531. 
is  the  same  in  whatever  direction  the  ray  be  incident,  viz.  in  a  given  ratio  to  the  velocity  before  incidence,  this  Law  of 
ratio  being  one  of  equality  in  the  case  of  reflexion.  velocities. 

Let  us  next  consider  the  direction  of  the  ray  after  flexure.     To  this  end  let  0  =  the  angle  made  by  its  path       55%. 

^  ^  Direction  of 

at  any  moment  with  the  perpendicular  to  the  surface,  then  will  sin  0  =  — — ,  putting  ds  for   ^dx3-^-  dy*,  the  'herayaft8r 

element  of  the   arc.     Now  if  we  integrate  the  equation  — — —  =  0   once    we   get  — —  =  constant  =  c,  and 

d  t  (it 

dx=cdt,  wherefore  win  0  =  — - — .     But  x  =  — ; — .therefore   sin  0  =  — .     Let  therefore  #„  and   0.  repre- 

d  s  d  t  v 

sent  the  initial  and  terminal  values  of  0,  or  the  angles  of  incidence  and  reflexion,  or  refraction  of  the  rectilinear 
DOrtionS  of  the  ray,  and  we  get  Coustancy 

of  ratio  of 
In/)    -  in,!  dn  n   —•     C  sines  of  in- 

blll  v~  —  f  dllu  all!  17.  . —  ->  .  .  . 

V  '         V'  cidence  and 

refraction. 

and  dividing  one  by  the  othei 

sin  60  V 

sin  0,     :     ~T~  ~  '*' 

That  is  to  say,  the  sines  of  incidence  and  refraction,  or  reflexion,  are  to  each  other  in  a  constant  ratio,  viz.  the 
inverse  ratio  of  the  velocities  of  the  ray  before  and  after  incidence. 

Thus  we  see  the  Newtonian  hypothesis  satisfies  the  fundamental  conditions  of  refraction  and  reflexion  without  533. 
entering  into  any  consideration  respecting  the  laws  of  the  refracting  and  reflecting  forces,  or  even  the  order  of 
their  superposition.  There  may  be  as  many  alternations  of  attraction  and  repulsion  as  we  please,  and  the 
reflected  or  refracted  ray  may  therefore,  prior  to  its  final  recess  from  the  surface,  make  any  variety  of  undulations ; 
all  that  is  required  is  the  extremely  rapid  decrease  of  the  function  Y  expressing  the  total  force  before  the  distance 
attains  a  sensible  magnitude. 

Hence  also,  V  and  V  being  the  velocities  before  and  after  incidence,  and  n  the  index  of  refraction,  we  have        534.    • 

V'rV::/,:!, 

which  shows,  that  when  a  ray  passes  from  a  rarer  medium  to  a  denser,  its  velocity  is  increased,  and  vice  versd. 

Moreover,  we  have  535. 

V'2  _  V2        /  V'  V  9  f—  Y  il 11  Refractive 

k  =  - =  (       -)   -  l  =  u'-  l  =  ~l V  power  of  a 

V°  V,  V  )  V«  medium. 

Now  if  we  suppose  the  form  of  the  function  Y  to  be  the  same  for  all  media,  and  that  they  differ  in  the  energy 
of  action  only  by  reason,  first,  of  a  greater  density,  owing  to  which  more  molecules  are  brought  within  the 
sphere  of  activity ;  and,  secondly,  by  reason  of  a  greater  or  less  affinity,  or  intensity  of  action  of  each  molecule, 
we  may  suppose  Y  to  be  represented  by  S  .  n .  (j)  (y~),  where  S  is  the  specific  gravity,  or  density,  n  the  intrinsic 
refractive  energy  of  the  medium,  and  0  (y)  a  function  absolutely  independent  of  the  peculiarities  of  the  medium, 
and  the  same  for  all  natural  bodies.  Hence  f—  Y  dy=  S  .n  .f  —  <t>(y)  d  y  =  S  .  n  .  constant  because 
f  —  0  (y)  dy  taken  from  y  =  -f-CDtoy=—  oc  will  now  be  an  absolute  numerical  constant.  We  have  then, 
according  to  this  doctrine, 

I*  —  1  V* 

n  =  -£— X   - 


2  .  constant 


If  p.  be  the  refractive  index  of  a  given  standard  ray  out  of  a  vacuum,  V  the  velocity  of  that  ray  in  vacuo  is  known, 
und  is  also  an  absolute  constant ;  so  that  n,  the  intrinsic  refractive  power  of  the  medium  is  proportional  to 
VOL.  iv.  3  M 


442  LIGHT. 

Light.      (refractive  index)  *  -  1  Part  III. 

__    -.  .  : .     Such  is  Newton  s  idea  of  the  refractive  power  of  a  medium  as  differing  from  its  ^^-..-^ 

specific  gravity 

.efractive  index.  It  rests,  however,  on  a  purely  hypothetical  assumption,  that  of  the  similarity  of  form  of  the 
law  of  force  for  all  media,  respecting  which  we  can  be  said  to  know  nothing  whatever.  For  a  table  of  its  values 
for  different  media,  see  the  Collection  of  Tables  at  the  end  of  this  Essay. 

536.  The  constancy  of  the  ratio  of  the  sines  of  incidence  and  refraction  has  here  been  derived  by  direct  integration 
Principle  of  of  the  fundamental  equations.     There  is,  however,  another  mode  of  deducing  this  important  law,  much  more 
»m*loaed°n  circl"tous>  it  's  true>  m  this  simple  case,  but  which  offers  peculiar  advantages  in  the  more  complicated  ones  of 

double  refraction ;  and  which,  therefore  we  shall  here  explain,  to  familiarize  the  reader  beforehand  with  its 
principle  and  mode  of  application.  It  consists  in  the  employment  of  what  is  called,  in  Dynamics,  the  principle 
of  least  action,  in  virtue  of  which  the  sum  of  each  element  of  the  trajectory  described  by  any  moving  molecule 
multiplied  by  the  velocity  of  its  description  (or  the  integral  fv  d  s)  is  a  minimum  when  taken  between  any  two 
fixed  points  in  the  trajectory.  The  trajectory  described  by  any  luminous  molecule  may  be  regarded  as  consisting 
of  two  rectilineal  portions,  or  hyperbolic  branches,  confounded  with  their  asymptotes,  and  one  curvilinear  one 
concentrated  in  a  space  of  insensible  magnitude,  a  physical  point.  Within  this  point  the  whole  operation  of  the 
flexure  of  the  ray,  however  complicated,  is  performed ;  and  here  the  velocity  is  variable.  In  the  branches  it  is 
uniform.  Suppose,  then,  A  and  B  to  be  any  two  fixed  points  in  these,  taken  us  points  of  departure  and  arrival 
of  a  ray,  and  let  C  be  the  point  in  the  surface  of  a  reflecting  or  refracting  medium  where  the  flexure  takes  place, 
and  suppose  A  C  =  S,  B  C  =  S'  and  let  a  be  the  excessively  minute  curvilinear  portion  of  the  ray  at  C,  and  v 
the  variable  velocity  with  which  it  is  described,  V  and  V  being  those  with  which  S  and  S'  are  described.  Then 
may  the  integral  fvd  s  be  resolved  into  the  three  portions  fV  dS  +  fvda  +f\'d  S'.  Of  these  the  second 
is  utterly  insensible,  by  reason  of  the  minuteness  of  a,  and  the  other  two,  since  V  and  V  are  constant,  become 
merely  V  .  S  +  V  .  S'. 

537.  The  position  of  C,  then,  with  respect  to  A  and  B,  will  be  determined  by  the  condition  V  .  S  +  V .  S1  =  a 
minimum,  A  and  B  being  supposed  fixed,  and  C  any  how  variable  on  the  surface.     Now,  in  the  case  before  us, 
V  the  velocity  of  the  light  before,  and  V  that  after  incidence,  are  both,  as  we  showed  in  Article  529  and  530, 
independent  of  the  direction   of  the  incident  and  reflected  or  refracted  rays,  or  of  the  position  of  C ;    and, 
therefore,  are  to  be  considered  as  absolute  constants  in  this  problem  of  minima,  which  is  thus  reduced  to  a 
simple  geometrical  question.     Given  A  and  B  to  find  C,  a  point  in  a  given  plane,  such  that  V  (=;  constant)  x 
A  C  +  V  (=  constant)  X  B  C  shall  be  a  minimum.     Nothing  is  easier  than  the  solution.     Put  a,  b,  c,  a',  b',  c1 
for  the  respective  coordinates  of  A  and  B,  and  x,  y,  o  for  that  of  C,  taking  the  given  plane  for  that  of  the  x,  y. 

Solution  of   Then  

the  geome-  V  .  S  +  V  .  S'  .  =  V  .  A/  (x  -  d)  »  +  (y  -  b)  *  +  c«~  +  V  .  -/  (X  —  a')4  +  (V  -  b')*~+~c* 

trical  pro- 
minimum     is  to  be  a  minimum  by  the  variation  of  x  and  y,  independent  of  each  other.     This  gives,  by  differentiation, 


and  this,  since  x  and  y  are  independent,  must  vanish,  whatever  values  are  assigned  to  d  x  and  dy,  therefore  we 
must  have  separately 

-J-  (a  -  x)  +  — ;-  (a'  -  x)  =  0 ;  -  (6  -  y)  +  -=r  W  -  y)  =  0.  (d) 

o  o  GO 

These  give,  respectively, 

S7  V         a'-x  .  S'  V        b'-y 

~S~~         V     '     a-x   '  S  V     ' 

by  equating  which  we  get 

or  multiplying  out  and  reducing 

b  —  b'          a  b'  —  b  a' 

d  —  t£  a  —  & 

and,  consequently, 

6  —  b1 

—  (a  -  x) 


This  equation  expresses,  that  the  two  portions  S  and  S'  of  the  ray  before  and  after  incidence  on   the  surface  at 
C  both  lie  in  one  plane,  and  that    this   plane  is  perpendicular  to   the  surface,  or  to  the  plane  of  the  coordi- 


nates f,  y 


aes  f,  y. 
Again,  if  we  resume  the  equations  (d~)  and  putting  them  under  the  form 

S'  (a  —  x) 
deduced.        Square  and  add  them  we  get 


Constancy  yi  V' 

of  the  ratio  S'  (a  —  x)  =  --  —  S  (a1  —  x)  ;  S'  (6  -  y)  =  --  —  (b'  -  y)  .  S. 

of  the  sines 


LIGHT.  443 


Now  if  we  put  0  for  the  angle  made  by  the  portion  S  with  a  perpendicular  to  the  surface,  or  the  angle  of  inci- 
dence of  the  ray,  and  0'  for  that  made  by  the  other  S'  with  the  same  perpendicular,  or  the  angle  of  emergence, 
we  shall  have  _  _  ___^ 

.  ^(g-.^  +  (6-y).  .,        V  (of  -,).  +  (,-,).• 


So  that  the  above  equation  is  equivalent  simply  to 

V 
sin  9  =  -—  .  sin  ff, 

which  is  the  same  with  the  result  before  obtained. 

The  principle  of  least  action,  then,  in  the  case  before  us,  has  enabled  us  to  dispense  with  one  integration  of     539. 
the  differential  equations  expressing  the  motion  of  the  luminous  molecule  ;   and  its  applicability  to  this  purpose  Advantages 
depends,  as  we  have  seen,  on  the  relation  between  V  and  V  ;   the  velocities  of  the  light,  before  and  after  inci-  afforded  by 
dence,  being  known.     This  relation  has  here  been   deduced  &  priori;  but  had  it  been  merely  known,  as  a'^  of1""", 
matter  of  fact,  a  conclusion  established  by  experiment,  it  would  not  be  on  that  account  the  less  applicable   to  action. 
the   same  purpose,  and   the  laws  of  refraction  and   reflexion  would  be  derivable  from  it  by  the  same  process. 
There  would,  however,  be  this  main  difference  ;   that,  in  the  latter  case,  we  should  have  no  occasion  to  employ 
the  differential  equations  at  all,  and  therefore  none  to  enter  into  any  consideration  of  the  forces  acting  on  the 
luminous  molecule,  or  their  mode  of  action.     The  principle  of  least  action  establishes,  independent  of,  and 
anterior  to,  all  particular  suppositions  as  to  the  forces  which  operate  the  flexure  of  the  ray,  (further  than  that  they 
are  functions  of  the  distances  from  their  origins  or  centres,)  an  analytical  relation  between  the  velocities  before 
and  after  incidence,  and  the  directions  of  its  direct  and  deviated  branches  ;    a  relation  nearly  as  general  as 
the  laws  of  dynamics  themselves,  and  expressive,  in  fact,  of  only  the  one  condition  above  mentioned.     And  this 
relation,  from  its  form,  enables  us,  whenever  the  relation  of  the  velocities  is  known,  to  determine  that  of   the 
directions  of  the  two  portions  of  the  ray,  and  vice  versd,  without  having  recourse  to  the  differential  equations  at 
all.     In  the  simple  case  before  us  this  may  seem  a  needless  refinement,  the  equations  being  so  simple.     It  is  Applicable 
otherwise,  however,  in  the  theory  of  double  refraction.     There  the  forces  in  action  are  altogether  unknown,  not  to  other 
only  in  respect  of  their  intensity,  but  of  their  directions  ;  and  so  far,  therefore,  from  being  able  in  that  theory  to  cases 
integrate  the  equations  of  the  ray's  motion,  we  cannot  even  express  them  analytically.     The  principle  we  are 
now  considering  is,  in  such  a  case,  all  the  ground  we  have  to  stand  upon  ;  and  has  been  ingeniously  and  ele- 
gantly applied   by  Laplace,  in  that  theory,  to  reduce  the  complicated  laws   of  double   refraction   under  the 
dominion  of  analysis. 

In  fact,  suppose  that  the  velocities  of  the  incident  and  deviated  portions  of  the  rays,  instead  of  being  the  same      540. 
in  every  direction,  varied  with  the  positions  of  these  portions  with  respect  to  the  surface  of  the  medium,  or  to  Mode  of  it* 
any  fixed  lines  or  axes  in  space.     Then  will  V  and  V,  instead  of  being  invariable,  be  represented  by  functions  application 
of  the  three  coordinates  of  the  point  C,  either  rectangular,  as  x,  y,  z  ;   or  polar,  as  0,  0,  and  7  ;  and  the  portions  in  Seneral- 
S  and  S'  of  the  rays  intercepted  between  A  and  B  respectively,  and  the  surface  at  C,  will,  in  like  manner,  be 
functions  of  the  same  coordinates.     So  that  the  condition 

V  .  S  +  V.  S'  =  a  minimum 

will  afford,  by  differentiation  and  putting  the  differential  equal  to  zero,  an  equation  of  the  form  "Lidx  +  Mdy 
+  N  d  z  =  0,  orLd0  +  Md0  +  Nd7=:0,  as  the  case  may  be.  The  equation  of  the  surface  also  being 
differentiated  affords  another  relation  of  the  same  kind  ;  and  these  being  the  only  conditions  to  which  the  diffe- 
rentials dx,  dy,  dz  are  subject,  we  may  eliminate  one,  and  put  the  coefficients  of  the  remaining  ones  separately 
equal  to  zero.  Thus  we  get  two  equations  between  the  coordinates,  which,  combined  with  that  of  the  surface, 
suffice  to  determine  them,  i.  e.  to  fix  the  point  C  at  which  the  ray  A  C  must  meet  the  surface,  in  order  that,  being 
there  deviated  by  the  action  of  the  medium,  it  may,  after  flexure,  proceed  to  B  ;  and  thus  the  problem  of 
reflexion  or  refraction  may  be  resolved  in  all  its  generality,  so  soon  as  the  nature  of  the  functions  V,  V  is 
known.  But  to  return  to  the  case  of  ordinary  reflexion  and  refraction,  from  which  this  is  a  digression. 

Let  us  consider,  a  little  more  in  detail,  what  may  be  conceived  to  happen  to  a  ray  at  the  confines  of  the  surface      541. 
of  a  medium.     We  may  suppose,  then,  that  there  exist  a  series  of  laminar  spaces,  or  strata,  within  which  the  Coarse  of  a 
attractive  and  repulsive  action  of  the  molecules  of  the  medium  alternately  predominate.     Of  these  there  may  ray.' 
be  any  number,  and  either  may  be  exterior  to  the  rest.     It  is,  in  fact,  the  assemblage  of  these  lamina;  which  is  ™flj?t**p 
to  be  regarded  as  the  surface  of  the  medium.     Suppose  now  a  ray  A  a  (fig.  1  19)  to  be  moving  towards  the  and  refratt- 
medium.     Its  course  will  be  rectilinear  up  to  a,  where  it  first  comes  within  the  action  of  the  medium.     If  the  i»g  medium 
first  stratum  into  which  it  enters  be  one  of  attraction,  its  course  will  be  bent  as  ab  into  a  curve  concave  towards  '^ceA- 
the  surface,  and  its  velocity  in  the  direction  perpendicular  to  the  surface  will  be  increased.     Arrived  at  b  let  the     *"" 
force  change  to  repulsive  ;  the  trajectory  will  have  at  b  a  point  of  contrary  flexure,  the  portion  b  c  within  this 
lamina  will  be  convex  to  the   surface,  and   the  velocity  towards  the  surface  will  be  diminished   in  the  whole 
progress  of  the  ray  through  it,  and  so  for  any  number  of  alternations.     Let  us  now  suppose,  that  in  passing 
through  any  repulsive  lamina,  as  C,  the  repulsion  should  be  so  strong,  or  the  original  velocity  of  approach  to  the 
surface  so  small,  as  that  the  whole  of  it  shall  be  destroyed.     In  this  case  the  ray  for  a  moment  will  be  moving 
as   at  C,  parallel  to  the  surface,  but  the  repulsive  force  continuing  its  action  will  turn  it  back;  and  the  forces 

3  M  2 


444 


LIGHT. 


Light. 


542. 

Motion  of  a 
ray  at  com- 
mon surface 
of  two 
media. 

543. 

Newtonian 
idea  of  a 
ntv  of  light 
as  composed 
i>l  a  succes- 
sion of 
molecules. 


Their 
distance 
inter  it. 


Their  ex- 
treme 
tenuity  il- 
lustrated. 


544. 
Partial  re- 
flexion ex- 
plained on 
.vewton's 
p.inciples. 


545. 
Reflexion 
more  co- 
pious at 
great  obli- 
quities. 


now  being  all  equal  to  what  they  were  before,  but  acting  in  a  contrary  direction  with  respect  to  the  motion  of     Part  III. 
the  molecule,  it  will  describe  a  portion  C  d'  c'  b'  a'  B  similar,  and  equal  to  the  portion  on  the  other  side  of  C.   ^—v-"-* 
This  is  the  case  of  reflexion.     But  suppose,  as  in  fig.  120,  the  ray  to  have  such  an  initial  velocity  of  approach, 
or  the  repulsive  forces  to  be  so  feeble,  compared  to  the  attractive,  that  before  its  whole  velocity  perpendicular  to 
the  surface  is  destroyed,  it  shall  have  passed  through   all  the  strata  of  attraction  and  repulsion,  and  entered 
the  region  where  the  forces  of  all  the  molecules  are  in  equilibrium,  as  at  e.     In  this  case   the  remainder  of  its 
course  will  be  rectilinear,  and  wholly  within  the  medium.     This  is  the  case  of  refraction.     In  both  cases,  it  is 
the  final  course  it  takes,  or  the  direction  of  the  asymptotic  branches  a1  B  or  e  B,  about  which  only  we  have  any 
knowledge  ;    of  the  number  and  nature   of  the  undulations  of  its  course  between  a  and  a',  or  e,  we  know 
nothing. 

The  whole  of  this  reasoning  applies  equally  to  the  motion  of  a  luminous  molecule  at  the  confines  of  two 
media,  as  at  the  surface  separating  one  medium  from  a  vacuum.  The  molecules  of  either  medium  being  sup- 
posed uniformly  distributed,  and  acting  equally  in  all  directions  around  them,  the  resultant  of  all  their  forces 
on  the  luminous  particle  must  be  perpendicular  to  the  common  surface,  which  is  all  that  is  required  in  the 
foregoing  theory. 

In  the  Corpuscular  doctrine,  a  ray  of  light  is  understood  to  mean  a  continued  succession  or  stream  of  mole- 
cules, all  moving  with  the  same  velocity  along  one  right  line,  and  following  each  other  close  enough  to  keep  the 
retina  in  a  constant  state  of  stimulus,  i.  e.  so  fast,  that  before  the  impression  produced  by  one  can  have  time  to 
subside  another  shall  arrive.  It  appears,  by  experiment,  that  to  produce  a  continued  sensation  of  light,  it  is 
sufficient  to  repeat  a  momentary  flash  about  8  or  10  times  in  a  second.  If  a  red-hot  coal  on  the  point  of  a 
burning  stick  be  whirled  round,  so  as  to  describe  a  circle,  and  the  velocity  of  rotation  be  greater  than  8  or  10 
circumferences  per  second,  the  eye  can  no  longer  distinguish  the  place  of  the  luminous  point  at  any  instant,  and 
the  whole  circle  appears  equally  bright  and  entire.  This  shows,  evidently,  that  the  sensation  excited  by  the  light 
falling  on  any  one  point  of  the  retina,  must  remain  almost  without  diminution  till  the  impression  is  repeated 
during  the  subsequent  revolution  of  the  luminary.  Now,  if  uninterrupted  vision  can  be  produced  by  momen- 
tary impressions,  repeated  at  intervals  so  distant  as  a  tenth  of  a  second,  it  is  easy  to  conceive  that  the  indivi- 
dual molecules  of  light  in  a  ray  will  not  require  to  follow  close  on  each  other  to  affect  our  organs  with  a 
continued  sense  of  light.  As  their  velocity  is  nearly  200,000  miles  per  second,  if  they  follow  each  other  at 
intervals  of  1000  miles  apart,  200  of  them  would  still  reach  our  retina  per  second,  in  every  ray.  This  conside- 
ration frees  us  from  all  difficulties  on  the  score  of  their  jostling,  or  disturbing  each  other  in  space,  and  allows  of 
infinite  rays  crossing  at  once  through  the  same  point  of  space  without  at  all  interfering  with  each  other,  espe- 
cially when  we  consider  the  minuteness  which  must  be  attributed  to  them,  that  (moving  with  such  swiftness) 
they  should  not  injure  our  organs.  If  a  molecule  of  light  weighed  but  a  single  grain,  its  inertia  would  equal 
that  of  a  cannon  ball  of  upwards  of  150  pounds  weight,  moving  at  the  rate  of  1000  feet  per  second.  What 
then  must  be  their  tenuity,  when  the  concentration  of  millions  upon  millions  of  them,  by  lenses  or  mirrors,  has 
never  been  found  to  produce  the  slightest  mechanical  effect  on  the  most  delicately  contrived  mechanism,  in 
experiments  made  expressly  to  detect  it.  (See  Mr.  Bennet's  Experiments,  Phil  Trans.  1792,  vol.  Ixxxii.  p.  87.) 

When  a  ray  of  light  falls  on  a  reflecting  or  refracting  surface,  since  all  its  molecules  move  with  equal  velocity 
and  are  incident  in  the  same  line,  it  would  seem  that  whatever  took  place  with  one  should  equally  happen  to 
all ;  and  that,  if  the  first  underwent  reflexion,  all  should  do  so  ;  while,  on  the  other  hand,  if  one  could  penetrate 
the  surface,  and  pursue  its  course  entirely  within  the  medium,  all  ought  to  do  the  same.  This,  however,  is 
contrary  to  experience;  as  whenever  a  ray  of  light  is  incident  on  the  exterior  surface  of  any  medium,  a  part 
only  is  reflected,  and  the  rest  enters  the  medium.  No  theory  can  be  satisfactory  which  does  not  render  a  good 
account  of  so  principal  a  fact.  The  Newtonian  doctrine  accounts  for  it  by  the  fits  of  easy  reflexion  and  trans- 
mission. To  understand  this  explanation  we  must  recur  to  the  ninth  postulate,  (Art.  526,)  and  suppose  two 
molecules  to  arrive  at  the  surface  under  the  same  incidence,  the  one  in  a  fit  of  easy  reflexion,  the  other  in  one 
of  easy  transmission.  The  former  will  then  be  more  affected  by  the  repulsive  forces,  the  latter  by  the  attractive 
of  the  molecules  of  the  medium  ;  and  hence  it  is  evident,  that  (he  one  may  be  reflected  under  circumstances  of 
incidence,  &c.  in  which  the  other  will  penetrate  the  surface  and  be  refracted.  Now  it  will  depend  entirely  on 
the  nature  of  the  medium,  and  the  initial  velocity  of  a  luminous  molecule  towards  it,  (which  is  as  the  cosine  of 
the  angle  of  incidence,)  whether  it  will  require  the  whole  exertion  of  its  repulsive  forces,  in  their  most  energetic 
manner,  to  destroy  that  velocity  and  produce  reflexion,  or  only  a  part  of  them.  In  the  former  case  only  such 
molecules  as  arrive  in  the  most  favourable  disposition  to  be  reflected,  or  in  the  most  intense  phase  of  a  fit  of 
easy  reflexion,  can  be  reflected.  In  the  latter,  such  as  arrive  in  less  favourable  dispositions,  or  in  less  intense 
phases  of  fits  of  reflexion,  may  be  reflected ;  and  if  the  repulsive  forces  of  the  medium  be  very  intense,  in 
comparison  with  the  attractive  ones,  or  if  the  obliquity  of  incidence  be  so  great  as  to  give  the  molecule  a  very 
small  velocity  perpendicular  to  the  surface,  even  those  molecules  which  arrive  in  the  less  energetic  phases  of  fits 
of  easy  transmission  may  still  be  unable  to  penetrate  the  strata  of  repulsion. 

Hence,  then,  we  see  that  the  proportion  of  the  molecules  of  a  ray  falling  on  the  surface  of  a  medium  in  every 
possible  state  or  phase  of  their  fits,  which  undergo  reflexion,  will  depend,  first,  on  the  nature  of  the  medium  on 
whose  surface  they  fall,  or  if  it  be  the  common  surface  of  two,  then  on  both ;  secondly,  on  the  angle  of  incidence. 
At  great  obliquities,  the  reflexion  will  be  more  copious ;  but  even  at  the  greatest,  when  the  incident  ray  just 
grazes  the  surface,  it  by  no  means  follows  that  every  molecule,  or  even  the  greater  part,  must  be  reflected.  Those 
which  arrive  in  the  most  favourable  phases  of  their  fits  of  transmission,  will  obey  the  influence  of  small  attrac- 
tive forces,  in  preference  to  strong  repulsive  ones ;  but  it  will  depend  entirely  on  the  nature  of  the  media  whether 
the  former  or  the  latter  shall  prevail,  the  fits  in  the  Newtonian  doctrine  being  conceived  only  to  dispose  the 
luminous  molecules,  other  circumstances  being  favourable,  to  reflexion  or  transmission  ;  to  exalt  the  forces  which 


LIGHT.  4i:» 

Light.      tend  to  produce  the  one  and  to  depress  those  which  act  in  favour  of  the  other,  but  not  to  determine,  absolutely,    Hart  III. 
**V~*  'ts  reflexion  or  transmission  under  all  circumstances.  T'T""" 

These  conclusions  are  verified  by  experience.     It  is  observed,  that  the  reflexion  from  the  surfaces  of  transparent       •' ' f>- 
(or  indeed  any)  media,  becomes  sensibly  more  copious  as  the  angle  of  incidence  increases  ;  but  at  the  external  ^™^ 
surface  of  a  single  medium  is  never  total,  or  nearly  total.     In  glass,  for  instance,  even  at  extreme  obliquities,  a  m-ent 
very  large  portion  of  the  light  still  enters  the  glass  and  undergoes  refraction.     In  opaque  media,  such  as  polished 
metals,  the  same  holds  good ;  the  reflexion  increases  in  vividness  as  the  incidence  increases,  but  never  becomes 
total,  or  nearly  so.   The  only  difference  is,  that  here  the  portion  which  penetrates  the  surface  is  instantly  absorbed 
and  stifled. 

The  phenomena  which  take  place  when  light  is  reflected  at  the  common  surface  of  two  media,  are  such  as  from       547. 
the  above  theory  we  might  be  led  to  expect, — with  the  addition,  however,  of  some  circumstances  which  lead  us  to  Reflex" 
limit  the  generality  of  our  assumptions,  and  tend  to  establish  a  relation  between  the  attractive  and  repulsive  j^f^™,,"" 
forces,  to  which  the  refraction  and  reflexion  of  light  are  supposed  to  be  owing.     For  it  is  found,  that  when  two  two  meaja. 
media  are  placed  in  perfect  contact,  (such  as  that  of  a  fluid  with  a  solid,  or  of  two  fluids  with  one  another,)  the 
intensity  of  reflexion  at  their  common  surface  is  always  less,  the  nearer  the  refractive  indices  of  the  media  approach 
to  equality ;  and  when  they  are  exactly  equal,  reflexion  ceases  altogether,  and  the  ray  pursues  its  course  in  the 
second  medium,   unchanged  either  in  direction,  velocity,  or  intensity.     It  is  evident,  from   this  fact,  which  is 
general,  that  the  reflective  or  refractive  forces,  in  all  media  of  equal  refractive  densities,  follow  exactly  the  same 
laws,  and  are  similarly  related  to  one  another  ;   and  that  in  media  unequally  refractive,  the  relation  between  the 
reflecting  and  refracting  forces  is  not  arbitrary,  but  that  the  one  is  dependent  on  the  other,  and  increases  and 
diminishes  with  it.     This  remarkable  circumstance  renders  the  supposition  made  in  Art.  535,  of  the  identity  of 
form  of  the  function  Y,  or  0  (y)  expressing  the  law  of  action  of  the  molecules  of  all  bodies  on  light  indif- 
ferently, less  improbable. 

To  show  experimentally  the  phenomena  in  question,  take  a  glass  prism,  or  thin  wedge  of  very  small  refracting      543. 
angle  (half  a  degree,  for  instance:  almost  any  fragment  of  plate  glass,  indeed,  will  do,  as  it  is  seldom  the  two  sides  Phenomena 
are  parallel,)  and  placing  it  conveniently  with  the  eye  close  to  it,  view  the  image  of  a  candle  reflected  from  the  exhibited 
exterior  of  the  face  next  the  eye.     This  will  be  seen  accompanied  at  a  little  distance  by  another  image,  reflected  exPer" 
internally  from  the  other  face,  and  the  two  images  will  be  nearly  of  equal  brightness,  if  the  incidence  be  not 
very  great.     Now,  apply  a  little  water,  or  a  wet  finger,  or,  still  better,  any  black  substance  wetted,  to  the  pos- 
terior face,  at  the  spot  where  the  internal  reflexion  takes  place,  and  the  second  image  will  immediately  lose  great 
part  of  its  brightness.     If  olive  oil  be  applied  instead  of  water,  the  defalcation  of  light  will  be  much  greater , 
and  if  the  substance  applied  be  pitch,  softened  by  heat,  so  as  to  make  it  adhere,  the  second  image  will  be  totally 
obliterated.     On  the  other  hand,  if  we  apply  substances  of  a  higher  refractive  power  than  glass,  the  second  image 
again  appears.     Thus,  with  oil  of  cassia  it  is  considerably  bright ;  with  sulphur,  it  cannot  be  distinguished  from 
that  reflected  at  the  first  surface ;   and  if  we  apply  mercury,  or  amalgam,  (as  in  a  silvered  looking-glass,)  the 
reflexion    at  the  common  surfact  of  the  glass  and  metal  is  much  more  vivid  than  that  reflected  from  the  glass 
alone. 

The  destruction  of  reflexion  at  the  common  surface  of  two  media  of  equal  refractive  powers  explains  many       549. 
curious  phenomena.     If  we  immerse  an  irregular  fragment  of  a  colourless  transparent  body  (as  crown  glass)  in  j 
a  colourless  fluid  of  precisely  equal  refractive  power,  it  disappears  altogether.     In  fact,  the  surface  being  only  '0' 
visible  by  the  rays  reflected  from  it ;  destroy  this  reflexion,  and  the  object  must  cease  to  be  seen,  unless  from  any  foregoing 
opacity  in  its  bubstnnce  reflecting  rays  from  its  interior,  which  is  not  here  contemplated.     Hence,  if  the  powder  principles. 
of  any  such  substance  De  moistened  with  a  fluid  of  the  same  refractive  density,  all  the  internal   and  external 
reflexions  at  the  surfaces  of  the  small  fragments  of  which  it  consists,  which,  blended  and  confused,  present  the 
general  appearance  of  a  white  opaque  mass,  will  be  destroyed,  and  the  powder  will  be  rendered  perfectly  trans-  Transpa- 
parent.     A  familiar  instance  of  this  nature  is  the  transparency  given  to  paper  by  moistening  it  with  water,  or,  rency  of 
still  better,  with  oil ;  paper  is  composed  of  an  infinity  of  minute  transparent,  or  nearly  transparent  fibres  of  a  oiled  paper 
ligneous  substance,  having  a  refractive  power  probably  not  very  different  from  some  of  the  more  refractive  oils. 
Its  whiteness  is  caused  by  the  confused  reflexion  of  the  incident  rays  at  all  possible  angles,  both  internally  and 
externally,  those  which  have  escaped  reflexion  at  one  fibre,  undergoing  it  among  those  beneath.     If  moistened 
with   any  liquid,  the  intensity  of  these  reflexions  is  weakened,  and  the  more  the  more  nearly  its  refractive  power 
approaches  to  that  of  the  paper  itself;    so  that  a  considerable  number  of  rays  find  their  way  through  it,  and 
emerge  at  the  posterior  surface.     The  transparency  acquired  by  the  hydrophane,  by  immersion  in  water,  is,  no 
doubt,  owing  to  this  cause ;  the  water  filling  up  the  minute   pores,  and  enfeebling  the  internal  reflexion ;   and 
Dr.  Brewster,  in  a  very  curious  and  interesting  Paper  on  the  tabasheer,  (a  siliceous  concretion  found  in  sugar- 
canes,  and  the  lowest  in  the  scale  of  refracting  powers  among  solids,)  has  explained  on  this  principle  a  number 
of  extraordinary  phenomena  exhibited  on  moistening  that  substance  with  various  liquids,  (see   Philosophical 
Transactions,  1819.) 

The  reasoning  of  Art.  529  applies,  it  is  evident,  equally  to  the  case  when  a  ray  is  reflected  from  the  interior      550. 
surface  of  a  dense  medium  placed  in  air,  and  when  from  the  exterior.     The  only  difference  is,  that  in  the  latter  Total 
case  the  reflexion  is  performed  by  the  action  of  repulsive,  and  in  the  former  by  that  of  attractive  forces.     The  "" 
course  of  a  ray  internally  reflected  may  be  conceived,  as  in  fig.  12  L  and  122  ;  and  the  reflexion  may  take  place  re 
in  any  of  the  attractive  regions,  or  laminae,  whether  within  or  without  the  true  surface,  i.  e.  the  last  layer  of 
molecules  which  constitute  the  medium.     There  is  one  case  of  internal  reflexion,  however,  too  remarkable  to  be 
passed  without  more  particular  notice.     It  is,  that  when  the  interior  angle  of  incidence  exceeds  the  limiting 

angle  whose  sine  is  — ,  (see  Art.  193   et  s?q. ;)  and   when,  as  we   there  stated,  as  a  result  of  experiment,  the 


446  LIGHT. 

Light.      internal  reflexion  is  total.     To  see  how  this  happens,  let  us  consider  a  ray  incident  exactly  at  this  angle,  and    P»rt  III. 

^— — \~™-"/  in  the  most  intense  phase  of  its  fit  of  transmission.     Then  will  it  be  refracted  ;  and,  since  the  angle  of  refraction  >-^— v--~ 
must  be  just  90°,  (by  reason  of  the  generality  of  the  demonstration  of  the  law  of  refraction  in  Art.  529,)  it 
will  emerge,  grazing  the  surface,  exactly  at  the  extreme  boundary  of  the  outermost  region  C  B,  (fig.  123,)  where 
all  sensible  action  ceases.     Its  initial  velocity  under  these  circumstances  in  the  direction  perpendicularly  to  the 
surface,  is  barely  sufficient  to   carry  it   up  to  this  extreme  limit,  where  it  is  quite  annihilated.     If,  then,  we 
conceive  another  ray,  also  incident  in  the  most  intense  phase  of  its  fit  of  transmission,  but  at  an  angle  more 
oblique  by  an  infinitely  small  quantity,  then,  since  its  initial  velocity  at  right  angles  to  the  surface  is  less,  it  will 
be  destroyed  before  it  has  quite  reached  this   limit,  and  the  ray  will  therefore  begin  to  move  parallel  to  the 
.surface,  just  within  the  last  limit  to  the  sphere  of  its  action. 
551.  Now  the  last  action  which  the  surface  exerts,  or  that  force  which  extends  to  the  greatest  distance  from  it, 

The  outer-    cannot  be  otherwise  than  attractive  ;  for,  first,  were  it  repulsive,  it  is  evident  that  no  ray  incident  externally  at 
re  an  extreme  incidence,  (i.  e.  approaching  indefinitely  to  90°,)  could  by  possibility  escape  reflexion ;  and,  secondly, 

necessarily    no  rav  on  tnat  supposition  could  emerge  from  within  the  medium,  without  having  at  its  emergence  an  obliquity 

attractive,  to  the  surface  greater  than  some  finite  angle,  the  last  action  of  the  medium  being  in  this  case  to  bend  it  outwards, 
both  which  consequences  are  contrary  to  fact.  Or  we  may  consider  the  point  thus,  Since  a  ray  incident  within, 
at  the  limiting  angle,  emerges,  if  it  emerge  at  all,  parallel  to  the  surface  ;  and  since  every  point  in  the  curve 
described  by  it  previous  to  the  instant  of  emergence  is  nearer  to  the  medium  than  the  line  of  its  ultimate 
direction,  it  is  geometrically  impossible  that  the  curvature  immediately  adjacent  to  the  point  of  emergence  should 
be  otherwise  than  concave  towards  the  medium  ;  and  must,  therefore,  of  necessity  be  produced  by  a  force  directed 
to  it,  i.  e.  an  attractive  one. 

Hence,  the  luminous  molecule  we  have  been  considering,  will  be  within  the  attractive  region  at  the  moment 
when  its  perpendicular  motion  is  destroyed ;  it  will,  therefore,  be  turned  inwards,  as  at  the  dotted  line  fig. 
123,  and  be  reflected.  A  fortiori,  therefore,  wili  every  molecule  incident  in  a  less  intense  phase  of  a  fit 
of  transmission,  or  in  one  of  reflexion,  as  well  as  every  one  incident  at  a  more  oblique  incidence,  i.  e.  with 
a  less  initial  perpendicular  velocity,  be  reflected.  Those  in  which  the  circumstances  are  most  favourable  to 
transmission  will  reach  the  exterior  attractive  region,  as  in  fig.  123.  Others  in  which  they  are  less  so  will  be 
reflected  in  some  intermediate  region,  as  in  fig.  122,  while  those  which  are  incident  at  extreme  internal  obli- 
quities, or  in  the  most  intense  phases  of  fits  of  reflexion,  will  have  their  courses  as  represented  in  fig.  121. 

553.  The  conclusion  at  which  we  have  arrived  in  the  last  Art.  that  the  attractive  force  of  a  medium  on  the  molecules 
llclre~  ,    of  light  extends  to  a  greater  distance  than  the  repulsive,  is,  as  we  have  seen,  a  necessary  consequence  of  dyna- 

liiiue  reflex-  m'ca'  principles  ;  and  so  far  from  being  in  opposition  to  Newton's  doctrine  of  reflexion,  as  has  been  said,  is  in 

ion  from       perfect  accordance  with  it.     Dr.  Brewster  has  been  led  to  the  same  conclusion  by  peculiar  considerations 

water.          grounded  on  his  experiments  on  the  law  of  polarization,  (JP/iil.  Trans.,  1815,  p.  133,)  and  has  applied  it  to 

explain  a  curious  fact  noticed  by  Bouguer,  viz.  that  although  water  be  much  less  reflective  than  glass  at  small 

incidences,  yet  at  great  ones  (as  87°£)  it  is  much  more  so.     Now,  supposing  the  light  to  have  undergone  the 

whole  effect  of  the  refracting  forces,  in  both  cases  before  it  suffers  reflexion,  its  incidence,  when  it  reaches  the 

region  of  the  repulsive  forces,  will  have  been  diminished  in  the  case  of  glass,  to  57°  44',  but  in  that  of  water 

only  to  61°  5',  and  therefore  being  incident  more  obliquely  on  the  water  it  ought  to  be  more  copiously  reflected. 

Whatever  we  may  think  of  the  validity  of  this  explanation,  it  is  certainly  ingenious,   and   the   fact  extremely 

remarkable,  and  deserving  of  all  attention. 

554.  To  see  the  phenomena  of  total  reflexion   to  the  best  advantage,  lay  down  a  right-angled  glass  prism  on   a 
Experiment  black  substance  close  to  a  window,  with  its  base  horizontal,  as  in  fig.  124,  and  apply  the  eye  close  to  the  side, 
showing  the  looking  downwards.     The  base  will  be  seen  divided  into  two   portions,  by  a  beautiful  coloured  arch   like  a 
nf  Tt"?™1  ra'nDOW  concave  to  the  eye,  the  portion  above  the* arch  being  extremely  brilliant  and  vivid,  and  giving  a  reflexion 
nflexion.     °f  a"   external  objects  no  way  to  be  distinguished  from  reality.      On   the  other   hand,  the  space    within  the 

concavity  of  the  bow  is  comparatively  sombre,  the  reflexion  of  the  clouds,  &c.  on  that  part  of  the  base  being 
much  less  vivid.  If,  instead  of  placing  it  on  a  black  body,  we  hold  it  in  the  hand,  and  place  a  candle  below  it, 
this  will  be  visible  ;  but  (wherever  placed)  will  always  appear  in  some  part  of  the  base  within  the  concavity  of 
the  bow.  Fig.  124  represents  the  course  of  the  rays  in  this  experiment,  E  being  the  eye,  NG,  OF,  PD  rays 
incident  through  the  farther  side  at  various  angles  of  obliquity  on  the  base,  and  reflected  to  the  eye  at  E,  of 
which  O  F  is  incident  precisely  at  the  limiting  angle.  It  is  obvious,  that  all  the  rays  towards  N  incident  on  that 
part  of  the  base  beyond  F  being  too  oblique  for  transmission  will  be  totally  reflected,  while  those  incident 
between  F  and  A,  being  less  oblique  than  is  required  for  total  reflexion,  will  be  only  partially  so,  a  portion 
escaping  through  the  base  in  the  direction  D  Q.  Again,  if  we  place  a  luminary  at  any  point  as  L  below  the 
base,  it  is  manifest  that  to  reach  the  eye,  a  ray  from  it  must  fall  between  A  and  F,  as  L  D,  and  that  no  ray 
falling  on  any  part  of  the  base  between  B  and  F  can  be  refracted  to  E. 

555.  The    coloured  arch  separating  the  region  of  total  from  that  of  partial   reflexion,  is  thus  explained.     For, 
Reflected     simplicity,  let  us  suppose  the  eye  within  the  medium,  (to  avoid  considering  the  reflexion  at   the  inclined  surface 

A  C  of  the  prism  ;)  and,  first,  considering  only  the  extreme  red  rays,  if  we  drop  a  perpendicular  from  the  eye  on 
the  base  of  the  prism,  and  make  this  the  axis  of  a  cone,  the  side  of  which  is  incline!  to  the  axis  at  the  angle 

whose   sine  is  ,  (or  the  limiting  angle  for  extreme  red  rays;)  and  if  we  conceive  such  ravs  to  emanate  in 

all  directions  from  the  eye,  then  all  which  fall  without  the  circular  base  of  this  cone  will  be  totally,  but  those 
within  only  partially  reflected.  Thus,  were  there  no  other  than  such  red  rays  of  this  precise  refrangibiiity,  the 


LIGHT.  447 

Light,      region  of  partial  reflexion  would  be  a  circle  whose  radius  =  height  of  the  eye  above  the  base  X  tangent  of  the     Part  III. 

angle  whose   sine  is  =  —  — .     In  like  manner,  the  radius  of  the  circular  space,  within  which  only  a 

/*         Vp*  -  l 

H  H 

partial  reflexion  of  violet  rays  takes  place,  is  — ^         __,    or  ,  being  less   than   the  value 


of  the  same  radius  for  the  red  rays.  Hence,  in  the  space  between  the  two  circles,  the  violet  rays  will  be  totally, 
and  the  red  only  partially  reflected ;  and,  therefore,  the  whole  of  this  space  will  have  an  excess  of  violet  light. 
A  similar  reasoning  holds  good  for  the  intermediate  rays;  and  the  shading  away  from  the  bright  space  without, 
to  the  comparatively  dark  one  within,  will,  in  consequence,  be  performed  by  the  abstraction  first  of  the  red,  next 
of  the  orange  rays,  and  so  on  through  the  spectrum,  leaving  a  residual  light,  which  continually  deviates  more 
and  more  from  white,  and  verges  to  blue.  If  now  we  suppose  each  ray  to  be  incident  in  the  contrary  direction 
so  as  to  be  reflected  to  the  eye  instead  of  emanating  from  it,  every  thing  will  equally  hold  good,  and  the  eye 
will  see  a  bright  space  without ;  separated  from  an  obscure  space  within  the  base  of  the  cone,  the  transition  from 
one  to  the  other  being  not  sudden,  but  marked  by  a  blue  border,  the  colour  of  which  is  more  lively  towards  the 
interior.  Now  such  is  the  fact,  with  one  difference,  however,  that  the  coloured  arch  appears  slightly  tinged 
with  pink  on  its  convex  side.  This,  as  it  is  incompatible  with  theory,  can  be  owing,  it  should  seem,  to  no 
cause  but  contrast ;  a  most  powerful  source  of  illusion  in  all  the  phenomena  of  colours,  and  of  which  this  is, 
perhaps,  one  of  the  most  striking  and  curious  instances.  Newton  (Optics,  part  ii.  exp.  16)  takes  no  notice  of 
this  part  of  the  phenomenon,  (which  was  first  observed  and  described  by  Sir  W.  Herschel,)  though  he  gives  the 
same  explanation  of  the  rest  with  that  here  set  down.  The  effect  of  refraction  at  the  side  B  A  of  the  prism 
will  somewhat  modify  the  figure  of  the  bow,  giving  it  a  tendency  to  a  conchoidal  form  at  great  obliquities  of 
the  emergent  rays. 

If  the  side  B  C  of  the  prism  be  covered  with  black  paper,  and  a  bright  scattered  light  be  thrown  on  the  base       555. 
from  below,  (as  from  an  emeried  glass  applied  with  its  rough  side  close  to  the  base,)  the  converse  of  the  above  Transmitted 
described  phenomena  will  be  seen.     A  totally  black  space  will  be  seen  beyond  F,  and  a  bright  one  within  it.    The  prismatic 
separation  being  marked  by  a  bow  of  a  vivid  red  colour,  graduating  through  orange  and  pale  yellow  into  white,  the  l)OW- 
red  being  outwards.     It  is  evident  that  this  phenomenon  is,  in  all  its  parts,  complementary  to  that  of  the  blue 
bow  seen  by  reflexion,  and  therefore  requires  no  more  particular  explanation.     It  should  be  noticed,  however,  that 
in  this  bow  no  appearance  of  blue  or  violet  within  its  concavity  is  ever  seen ;    so  that  the  effect  which  we 
have  above  attributed  to  contrast  in  the  reflected  bow  has  nothing  corresponding  to  it  in  the  transmitted  one. 

The  intensity  and  regularity  of  reflexion   at   the   external  surface  of   a  medium,  is  found   to    depend    not      557. 
merely  on  the  nature  of  the  medium,  but  very  essentially  on  the  degree  of  smoothness  and  polish  of  its  surface.  Reflexion 
But  it  may  reasonably  be  asked,  how  any  regular  reflexion  can  take  place  on  a  surface  polished  by  art,  when  we  jrtjfjcjaiiyS 
recollect  that  the  process  of  polishing  is,  in  fact,  nothing  more  than  grinding  down  large  asperities  into  smaller  poijshed 
ones  by  the  use  of  hard  gritty  powders,  which,  whatever  degree  of  mechanical  comminution  we  may  give  them,  explained. 
are  yet  vast  masses,  in  comparison  with  the  ultimate  molecules  of  matter,  and  their  action  can  only  be  considered 
as  an  irregular  tearing  up  by  the  roots  of  every  projection  which  may  occur  in  the  surface.     So  that,  in  fact,  a 
surface  artificially  polished  must  bear  somewhat  of  the  same  kind  of  relation  to  the  surface  of  a  liquid,  or  a 
crystal,  that  a  ploughed  field  does  to  the  most  delicately  polished  mirror,  the  work  of  human  hands.     Now  to 
this  question  the  Newtonian  doctrine  furnishes  an  answer  quite  satisfactory.     Were  the  reflexion  of  light  per- 
formed by  actual  impact  of  its  molecules  upon  those  of  the   reflecting  medium,  no  regular  ordinary  reflexipn 
could  ever  take  place  at  all,  as  it  would   depend  entirely  on  the   shape  of  the  molecules,  or  asperities  of  the  Light  not 
surface,  and  the  inclinations  of  their  surfaces  to  the  general  surface  of  the  medium  at  the  point  of  incidence,  j^f,'*^  ' 
what  should  be  the  direction  ultimately  taken  by  each  particular  ray.     Now  these  must  vary  in  every  possible  pact  on 
manner  in  uncrystallized  bodies,  so  that  in  reflexion  from  the  surfaces  of  these  the  light  would  be  uniformly  scat-  bodies. 
tered  in  every  direction.     On    the  other   hand,   in  crystallized  media,  each  molecule  presenting  only  a  limited 
number  of   strictly  plane  surfaces,  and  the  corresponding  faces  of  all  being  mathematically  parallel,  reflexion 
would  indeed  be  regular  ;  but  its  direction  would  be  regulated  only  by  that  of  the  incident  ray  and  the  position 
of  certain  fixed  lines  within  the  crystal ;    and  would  be  quite  independent  of  either  the  smoothness  or  the 
inclination  of  the  polished  surfaces  of  it,  whether  natural  or  artificial  ;  add  to  which,  that  instead  of  the  reflected 
pencil  of  rays  being  single,  it  would  in  most  cases  be  multiple.     All  these  consequences  are  so  contrary  to  fact,  But  by 
that  it  is  evident  we  must  suppose  the  distance  to  which  the  forces  producing  reflexion  extend  much  greater  °' 
not  only  than  the  size  of,  or  interval  between  individual  molecules,  but  even  greater  than  the  minute  inequalities 
or  furrows  in  the  artificially  polished  surfaces  of  media.     Granting  this,  the  difficulty  vanishes ;  for  the  average 
action  of  many  molecules,  or  many  corrugations,  will   present  an  uniformity,  while  individually  they  may  offer 
the  greatest,  diversity.     To  illustrate  this,  we  need  only  cast  our  eyes  on  fig.  125,  where   A  B   represents  the 
rough  surface  of  a  medium,  and  A  C  the  radius  of  one  of  the  spheres  of  attraction,  or  repulsive  activity  of  a 
single  molecule  A.     Conceiving  now  the  summits  of  all  the  elevations  a,  b,  c,  d  to  lie  in  a  plane,  let  spheres  be 
described  with  their  centres  equal  to  A  C.     Then  their  intersections  will  generate  a  kind  of  mamillated  surface 
a  ft  <•{  S,  which,  however,  if  the  radii  of  the  spheres  be  at  all  considerable  with  respect  to  the  distances  of  their 
centres,  will  approach  exceedingly  near  to  a  mathematical  plane,  infinitely  more  so  than  the  surface  A  B  need  be 
supposed.     Hence,  a  ray  of  light  impinging  on  the  medium  will  come  within  the  sphere  of  its  action  not  at  an 
irregular  surface,  but  nearly   at  a  plane  one  ;  and  the  resultant  action  of  all  the  molecules  in  action,  supposing 
them  distributed  with  uniformity  over  A  B,  will  be  perpendicular  to  this  surface.     The  same  will  hold  good  of 
the  layer  of  molecules  (however  interrupted)  immediately  under  the  summits  6,  c,  d,  &c.,  and  ot  all  the  other 


448  L  I  G  II  T. 

I.ii_'ht.      layers  into  which  the  whole  surface  can  be  divided.     So  that  the  essential  conditions  on  which  the  Newtonian     F'«a  ill 
•_— -v~»<'  doctrine  of  reflexion  and  refraction  reposes,  (viz.  equality  of  force  at  equal  distances  from  the  general  level  of1— ~v-"- 

the  surface,  and  the  perpendicularity  of  its  direction  to  that  level,)  still  obtain. 

558.  It  is  evident  that  the  inequalities  in  the  mamillary  surfaces  above  described  will  become  more  considerable  as 

Oblique  their  radii  are  diminished,  or  as  the  interval  of  their  centres  is  greater,  and  in  proportion  will  the  regularity  of 
regular  re-  reflexion  and  refraction  be  interrupted.  Hence  too  it  follows,  that  the  more  oblique  the  incidence  of  the  ray,  the 
rough  greater  maybe  the  roughness  of  t  lie  surface  which  will  give  a  regular  reflexion;  and  this  is  perfectly  con- 

sartaces.  sonant  to  fact,  as  may  be  easily  tried  with  a  piece  of  emeried  glass,  which,  although  so  rough  as  to  give  no 
regular  image  at  a  perpendicular  incidence,  will  yet  give  a  pretty  distinct  one  at  great  obliquities.  The 
reasons  are,  first,  that  a  very  oblique  ray  will  not  require  to  penetrate  so  far  within  the  sphere  of  repulsion,  to 
have  its  motion  perpendicular  to  the  surface  destroyed  ;  and,  secondly,  that  it  cannot  pass  between  two  conti- 
guous elevations  or  depressions  of  the  imaginary  surface  n  ft  7  £,  but  by  reason  of  its  obliquity  must  traverse 
several  of  them,  and  thus  undergo  a  more  regular  average  exertion  of  the  forces  of  the  medium. 

5"i9.  Thus  the  reflexion  of  light  is  explained  on  the  Newtonian  doctrine.     But  it  may  still  be  asked,  how  refraction 

Regular        at  a  surface  artificially  polished  can  ever  be  regular.     In  reflexion,  the  ray  never  reaches  the  asperities  o  '  the 
retraction      surface  ;   it  undergoes  their  average  action,  equalized  by  distance,  and  mutually  compensated.     In  retraction,  it 
•trtific Ml"    's  otnerw'se-     Here  the  rays  must  actually  traverse  the  surface,  and  must  therefore  actually  pass  through  all 
polished.      'ts  inequalities  at  every  possible  angle  of  obliquity.     The  answer  to  this  is  equally  plain.     Neither  refraction  nor 
reflexion  are  performed  close  to  the  surface,  either  wholly,  or  in   great  part.     The  greater  part  by  far  of  the 
flexure  of  the  ray  is  performed  (either  internally  or  externally)  at  a  distance,  out  of  the  reach  of  these  irregu- 
larities, and  by  the  action  of  a  much  more  considerable  thickness  of  the  medium  than  is   occupied   by  them. 
Their  action  must  be  compared  to  the  effect  of  mountains  on  the  earth's  surface  in  disturbing  the  general  force 
of  gravity.     A  stone  let  fall  close  to  one  of  them,  from  a  moderate  height,  follows  not  the  true  vertical  but  the 
direction  of  the  deviated  plumbline,  which  is  sensibly  different.     Wh  Teas,  if  let  fall  from  the  moon  to  the  earth's 
centre,  it  would  pass  among  them,  were  they  greater  a  tiiousand  fold  than  they  are,  without  experiencing  any 
sensible  perturbation  or  change  of  direction  in  their  neighbourhood. 

560.  In  fact,  however,  no  regular  refraction  can  be  obtained  from  surfaces  sensibly  rough,  at  all  comparable  to  the 
regularity  of  their  reflexion.     This  may  arise  from  the  impossibility  of  a  refracted  ray  penetrating  the  surface 
at  a  sufficient  degree  of  obliquity.     It  is,  however,  a  remarkable  fact,  that  the  regular  internal  reflexion  from  a 
roughened  surface,  even  at  extreme  obliquities,  is  scarcely  sensible,  even  in  cases  where  the  external  reflexion  at 
the  same  obliquities  is  perfectly  regular  and  copious.     T"\is   would   seem  to  indicate,   that  the   forces  which 
operate  the  external  reflexion  of  a  ray  exert  their  energy  wholly  without  the  medium. 

561.  Whatever  be  the  forces  by  which   bodies   reflect  and  refract  light,  one  thing  is   certain,  that  they  must  be 
Intensity  of  incomparably  more  energetic   than   the  force  of  gravity.     The   attraction   of  the   earth  on  a  particle  near  its 

forces  surface  produces  a  deflexion  of  only  about  16  feet  in  a  secor.  ;  and,  therefore,  in  a  molecule  moving  with  the 
1  velocity  of  light,  would  cause  a  curvature,  or  change  of  direction,  absolutely  insensible  in  that  time.  In  fact, 
u'e  must  consider,  first,  that  the  time  during  which  the  whole  action  of  the  medium  takes  place,  is  only  that 
within  which  light  traverses  the  diameter  of  the  sphere  of  sensible  action  of  its  molecules  at  the  surface.  To 
allow  so  much  as  a  thousandth  of  an  inch  for  this  space  is  beyond  all  probability,  and  this  interval  is  tra- 
versed by  light  in  the  — -  part  of  a  second.  Now,  if  we  suppose  the  deviation  produced 

1  2,  6 1 2,000,000,000 

by  refraction  to  be  30°,  (a  case  which  frequently  happens,)  and  to  be  produced  by  a  uniform  force  acting 
during  a  whole  second;  since  this  is  equivalent  to  a  linear  deflexion  of  200,000  miles  X  sin  30°,  or  of  100,000 
miles  =  33,000,000  x  16  feet,  such  a  force  must  exceed  gravity  on  the  earth's  surface  33,000,000  times. 
But,  in  fact,  the  whole  effect  being  produced  not  in  one  second,  but  in  the  small  fraction  of  it  above  mentioned, 
the  intensity  of  the  force  operating  it  (see  MECHANICS)  must  be  greater  in  the  ratio  of  the  square  of  one 
second  to  the  square  of  that  fraction ;  so  that  the  least  improbable  supposition  we  can  make  gives  a  mean 
force  equal  to  4,969,126,272  X  1094  times  that  of  terrestrial  gravity.  But  in  addition  to  this  estimate  already 
so  enormous,  we  have  to  consider  that  gravity  on  the  earth's  surface  is  the  resultant  attraction  of  its  whole 
mass,  whereas  the  force  deflecting  light  is  that  of  only  those  molecules  immediately  adjoining  to  it,  and  within 
the  sphere  of  the  deflecting  forces.  Now  a  sphere  of  TlrW  of  an  inch  diameter,  and  of  the  mean  density  of 

1  inch 

the  earth,  would  exert  at  its  surface  a  gravitating  force  only  TT,VTT  X  — — — rr-r —  .  •  of  ordinary  gra- 
vity, so  that  the  actual  intensity  of  the  force  exerted  by  the  molecules  concerned  cannot  be  less  than 

1000  X  earth  s  diameter    ^_  46)352)ooo,000)   times  the    above   enormous    number,   or  upw.irds  of  2  x   It)44 

1  inch 

when  compared  with  the  ordinary  intensity  of  the  gravitating  power  of  matter.  Such  are  the  energies  concerned 
in  the  phenomena  of  light  on  the  Newtonian  doctrine.  In  the  undulatory  hypothesis,  numbers  not  less  immense 
will  occur ;  nor  is  there  any  mode  of  conceiving  the  subject  which  does  not  call  upon  us  to  admit  the  exertion 
of  mechanical  forces  which  may  well  be  termed  infinite. 

r  fi>2  Dr.  Wollaston  has  proposed  the  observation  of  the  angle  at  which  total  reflexion  first  takes  place  at  the 

common  surface  of  two  media,  the  index  of  refraction  of  one  of  which  is  known,  as  a  means  of  determining 
that  of  the  other;  and,  in  the  Philosophical  Transactions  for  1802,  has  described  an  ingenious  apparatus  which 
gives  a  measure  of  the  index  required  almost  by  inspection.  If  we  lay  any  object  under  the  base  of  a  prism 


LIGHT.  449 

of  fl|nt  g.]ass  w;tn  a;r  a]one  interposed,  the  internal   angle  of  incidence  at  which  the  visual  ray  begins   to  be   Part  HI- 
totally  reflected,  and  at  which  of  course  the  object  ceases  to  be  seen  by  refraction  is  about  39°  10' ;  but  when  '-— ~v~ • — ' 
the  object  has  been  dipped  in  water,  and  brought  into  contact  with  the  glass,  it  continues  visible  (while  the  eye  Dr' iw°"as- 
is  depressed)  by  means  of  the  greater  refractive  power  of  the  water,  as  far  as  57^°  of  incidence.     When  any  t[)oll  of 
kind  of  oil,  or  any  resinous  cement,  is  interposed,  this  angle  is  still  greater,  according  to  the  refractive  power  of  determining 
the  medium  employed ;  and  by  cements  that  refract  more  strongly  than  the  glass,  the  object  may  be  seen  through  refractive 
the  prism  at  whatever  angle  of  incidence  it  is  viewed.     All  that  is  requisite,  then,  to   determine  the  refractive  P°wer3- 
index  of  any  body  less  refractive  than  glass,  is  to  bring  the  substance  to  be  examined  in  optical  contact  with 
the  base  of  a  prism,  and  to  depress  the  eye  (or  increase  the  angle  of  incidence)  till  it  ceases  to  be  seen  as  a 
dark  spot  on  the  silvery  reflexion  of  the  sky  on  the  rest  of  the  base.     With  fluids  and  soft  solids,  or  fusible 
ones,  the  requisite  contact  is  easily  obtained  ;  but  with  solids,  they  must  be  brought  to  smooth  surfaces,  and 
applied  to  the  base  by  the  intervention  of  some  fluid  or  cement  of  higher  refractive  power  than  the  glass,  which 
(since  the  surfaces  of  the  interposed  stratum  are  parallel)  will  produce  no  change  in  the  total  deviation  of  a 
ray  passing  through  it,  and  therefore  no  error  in  the  result.     By  this  method,  opaque  as  well  as  transparent 
substances  may  be  examined,  or  bodies  of  unhomogeneous  density,  as  the  crystalline  lens  of  the  eye.     It  may 
seem  paradoxical  to  speak  of  the  refractive  power  of  an  opaque  body  ;  but  it  will  be  remembered,  that  opacity 
is   merely  a  consequence  of  intense  absorbent  power,  and  that  before  a  ray  can  be  absorbed,  it  must  enter  the 
medium,  and  of  course  obey  the  laws  of  refraction  at  its  surface.     By  this  method,  Dr.  Wollaston  has  determined 
the  refractions  of  a  great  variety  of  bodies  ;  but  Dr.  Brewster  remarks,  that  the  method  must  be  liable  to  some 
source  of  inaccuracy,  which  renders  it  unsafe  to  trust  entirely  to  it  in  practice.     Dr.  Young  has  remarked,  that 
the  index  of  refraction  given  by  it,  belongs  in  strictness  to  the  extreme  red  rays. 

§  II.   General  Statement  of  the  Undulatory  Theory  of  Light. 

The  undulatory  theory,  among  whose  chief  supporters  we  have  to  number  Huygens,  Descartes,  Hooke,  and      563. 
Euler,  and,  in  later  times,  the  illustrious  names  of  Young  and  Fresnel,  who  have  applied  it  with   singular 
success  and  ingenuity  to  the  explanation  of  those  classes  of  phenomena  which  present  the  greatest  difficulties 
to  the  Corpuscular  doctrine,  requires  the  admission  of  the  following  hypotheses  or  postulata : 

1.  That  an  excessively  rare,  subtle,  and  elastic  medium,  or  ether,  as  it  is  called,  fills  all  space,  and  pervades  Postulata 
all  material  bodies,  occupying  the  intervals  between  their  molecules;  and,  either  by  passing  freely  among  them,  in  the 
or,  by  its  extreme  rarity,  offering  no  resistance  to  the  motions  of  the  earth,  the  planets,  or  comets  in  their  orbits,  sFtem  of 
appreciable  by  the  most  delicate  astronomical  observations ;  and  having  inertia,  but  not  gravity. 

2.  That  the  molecules  of  the  ether  are  susceptible  of  being  set  in  motion  by  the  agitation  of  the  particles  of 
ponderable  matter,  and  that  when  any  one  is  thus  set  in  motion  it  communicates  a  similar  motion  to  those 
adjacent  to  it ;    and  thus  the  motion  is  propagated  further  and  further  in  all  directions,  according  to  the  same 
mechanical  laws  which  regulate  the  propagation  of  undulations  in  other  elastic  media,  as  air,  water,  or  solids, 
according  to  their  respective  constitutions. 

3.  That  in   the  interior  of  refracting  media  the  ether  exists  in   a  state  of  less  elasticity,  compared  with  its 
density,  than  in  vacuo,  (i.  e.  in  space  empty  of  all  other  matter ;)  and  that  the  more  refractive  the  medium,  the 
less,  relatively  speaking,  is  the  elasticity  of  the  ether  in  its  interior. 

4.  That  vibrations  communicated  to  the  ether  in  free  space  are  propagated  through  refractive  media  by  means 
of  the  ether  in  their  interior,  but  with  a  velocity  corresponding  to  its  inferior  degree  of  elasticity. 

5.  That  when  regular  vibratory  motions  of  a  proper  kind  are  propagated   through  the  ether,  and,  passing 
through   our  eyes,  reach  and  agitate  the  nerves  of  our  retina,  they  produce  in  us  the  sensation  of  light,  in  a 
manner  bearing  a  more  or  less  close  analogy  to  that  in  which  the  vibrations  of  the  air  affect  our  auditory  nerves 
with  that  of  sound. 

6.  That  as,  in  the  doctrine  of  sound,  the  frequency  of  the  aerial  pulses,  or  the  number  of  excursions  to  and 
fro  from  its  point  of  rest  made  by  each  molecule  of  the  air,  determines  the  pitch,  or   note,  so,  in  the  theory  of 
light,  the  frequency  of  the  pulses,  or  number  of  impulses  made  on  our  nerves  in  a  given  time  by  the  ethereal 
molecules  next  in  contact  with  them,  determines  the  colour  of  the  light ;   and  that  as  the  absolute  extent  of  the 
motion  to  and  fro  of  the  particles  of  air  determine  the  loudness  of  the  sound,  so  the  amplitude,  or  extent  of  the 
excursions  of  the  ethereal   molecules  from   their  points  of  rest,  determine  the   brightness  or  intensity  of  the 
light. 

The  application  of  these  postulates  to  the  explanation  of  the  phenomena  of  light,  presumes  an  acquaintance      554 
with  the  theory  of  the  propagation  of  motion  through  elastic  media.     This  we  shall  assume,  referring  to  our  The  vtlo- 
article  on   sound  for  the  demonstration  of  all  the  properties  and  laws  of  motions   so  propagated,  as  we  shall  c|ty  of  a" 
have  occasion  to  employ.     One  of  the  principal  of  these  is,  that  supposing  the  elastic  medium  uniform   and  und"'atlon 
homogeneous,  all  motions  of  whatever  kind  are  propagated  through  it  in  all   directions  with  one  and  the  same  cqual' 
uniform  velocity,  a  velocity  depending  solely  on  the  elasticity  of  the  medium  us  compared  with  its  inertia,  and 
bearing  no  relation  to  the  greatness  or  smallness,  regularity  or  irregularity  of  the  original  disturbance.     Thus, 
while  the  intensity  of  light,  like  that  of  sound,  diminishes  as  the  distance  from  its  origin  increases,  its  velocity 
remains  invariable ,  and  thus,  too,  as  sounds  of  every  pitch,  so  light  of  every  colour,  travels  with  one  and  the 
same  velocity,  either  in  vacuo,  or  in  a  homogeneous  medium. 

Now  here  arises,  in  limine,  a  great  difficulty;   and  it  must  not  be  dissembled,  that  it  is  impossible  to  look  on 
VOL.  iv.  3  N 


450 


LIGHT. 


Light. 

Objection 
from  the 
pnenomena 
•  I  disper- 
sion. 


.566. 

Objection 
from  the 
rectilinear 
propagation 
of  light 
answered. 


667. 

Mode  in 
which  the 
rellna  is 
excited  by 
vil  ratio  s 
ui    ethe 


it  in  any  other  light  than  as  a  most  formidable  objection  to  the  undulatory  doctrine.  It  will  be  shown  presently  ^ 
that  the  deviation  of  light  by  refraction  is  a  consequence  of  the  difference  of  its  velocities  within  and  without 
the  refracting  medium,  and  that  when  these  velocities  are  given  the  amount  of  deviation  is  also  given.  Hence 
it  would  appear  to  follow  unavoidably,  that  rays  of  all  colours  must  be  in  all  cases  equally  refracted ;  and  that, 
therefore,  there  could  exist  no  such  phenomenon  as  dispersion.  Dr.  Young  has  attempted  to  gloss  over  this 
difficulty,  by  calling  in  to  his  assistance  the  vibrations  of  the  ponderable  matter  of  the  refracting  medium  itself, 
as  modifying  the  velocity  of  the  ethereal  undulations  within  it,  and  that  differently  according  to  their  frequency, 
and  thus  producing  a  difference  in  the  velocity  of  propagation  of  the  different  colours  ;  but  to  us  it  appears  with 
more  ingenuity  than  success.  We  hold  it  better  to  state  it  at  once  in  its  broadest  terms,  and  call  on  the  reader 
to  suspend  his  condemnation  of  the  doctrine  for  what  it  apparently  will  not  explain,  till  he  has  become 
acquainted  with  the  immense  variety  and  complication  of  the  phenomena  which  it  will.  The  fact  is,  that 
neither  the  corpuscular  nor  the  undulatory,  nor  any  other  system  which  has  yet  been  devised,  will  furnish  that 
complete  and  satisfactory  explanation  of  all  the  phenomena  of  ligh't  which  is  desirable.  Certain  admissions 
must  be  made  at  every  step,  as  to  modes  of  mechanical  action,  where  we  are  in  total  ignorance  of  the  acting 
forces ;  and  we  are  called  on,  where  reasoning  fails  us,  occasionally  fur  an  exercise  of  faith.  Still,  if  we  regard 
hypotheses  and  theories  as  no  other  way  valuable  than  as  means  of  classifying  and  grouping  together  pheno- 
mena, and  of  referring  facts  to  laws  which,  though  possibly  empirical,  are  yet,  so  far  as  they  are  so,  correct 
representations  of  nature,  and  as  such  must  be  deducible  from  real  primary  laws,  whenever  they  shall  be  disco- 
vered, we  cannot  but  admit  their  importance.  The  undulatory  system  especially  is  necessarily  liable  to  consi- 
derable obscurities  ;  as  the  doctrine  of  the  propagation  of  motion  through  elastic  media  is  one  of  the  most 
abstruse  and  difficult  branches  of  mathematical  inquiry,  and  we  are  therefore  perpetually  driven  to  indirect  and 
analogical  reasoning,  from  the  utter  hopelesness  of  overcoming  the  mere  mathematical  difficulties  inherent  in 
the  subject  when  attacked  directly. 

It  is  thus  that  we  are  encountered  at  the  very  outset  of  its  application  with  another  objection,  which,  in  the 
eyes  of  Newton,  appeared  decisive  against  its  admission,  but  which  has  since  been,  in  a  considerable  degree, 
overcome.  How  is  it  that  shadows  exist.  Sounds  make  their  way  freely  round  a  corner, — why  does  not  light 
do  so  ?  A  vibration  propagated  from  a  centre  in  an  elastic  medium,  and  intercepted  by  an  immovable  obstacle 
having  a  small  orifice,  ought  to  spread  itself,  it  is  said,  from  this  orifice  beyond  the  screen  as  from  a  new  centre, 
and  fill  the  space  beyond  with  undulations  propagated  from  it  in  every  direction.  Thus,  as  in  Acoustics,  the 
orifice  is  heard  as  a  new  source  of  sound  ;  so,  in  Optics,  it  ought  to  be  seen  in  all  directions  as  a  new  luminary. 
To  this  the  answer  is,  first,  that  it  is  not  demonstrable  that  a  vibratory  motion  communicated  to  one  particle  of 
an  elastic  medium  is  propagated  with  equal  intensity  to  ever}'  surrounding  molecule  in  whatever  direction 
situated  with  respect  to  the  line  of  its  motion,  though  it  is  with  equal  rapidity  ;  and  therefore  that  we  have  no 
reason  to  presume,  &  priori,  but  rather  the  contrary,  that  the  motions  of  the  vibrating  particles  at  the  orifice 
should  be  propagated  laterally  with  equal  intensity  in  all  directions ;  secondly,  that  it  is  not  true,  in  fact,  that 
sounds  are  propagated  round  the  corner  of  an  obstacle  with  the  same,  intensity  as  in  their  original  direction,  as 
any  one  may  convince  himself  by  the  following  simple  experiment.  Take  a  common  tuning  fork,  and,  holding 
it  (when  set  in  vibration)  about  three  or  four  inches  from  the  ear,  with  its  flat  side  towards  it,  when  its  sound 
is  distinctly  heard,  let  a  strip  of  card,  somewhat  longer  than  the  flat  of  the  tuning  fork,  be  interposed,  at  about 
half  an  inch  from  the  fork.  The  sound  will  be  almost  entirely  intercepted  by  it ;  and  if  the  card  be  alternately 
removed  and  replaced  in  pretty  quick  succession,  alternations  of  sound  and  silence  will  be  perceived ;  proving 
that  the  undulations  of  the  air  are  by  no  means  propagated  with  equal  intensity  by  the  circuitous  route  round 
the  edge  of  the  card,  as  by  the  direct  one.  Indeed  any  one  has  only,  to  be  convinced  of  the  fact,  to  attend  to 
the  sound  of  a  carriage  in  the  act  of  turning  a  corner  from  the  street  in  which  he  happens  to  be  to  an  adjoining 
one  ;  to  which  we  may  add,  that,  even  when  there  is  no  obstacle  in  the  way,  sounds  are  by  no  means  equally 
audible  in  all  directions  from  the  sounding  body,  as  any  one  may  convince  himself  by  holding  a  vibrating  tuning 
fork,  or  pitchpipe,  near  his  ear,  and  turning  it  quickly  on  its  axis.  This  last  phenomenon  was  first  noticed,  we 
believe,  by  Dr.  Young,  (Phil.  Trans.,  1802,  p.  25,)  and  since  more  fully  described  (in  Schweiggers  Jahrbuch, 
1826)  by  M.  Weber.  Now  if  there  be  any  inequality  at  all  in  the  intensity  of  the  direct  and  lateral  propa- 
gation of  undulations  in  a  medium,  it  must  arise  from  the  constitution  of  the  medium,  and  the  proportion  of 
the  amplitude  of  the  excursions  of  the  vibrating  particles  to  their  distance  from  each  other  ;  and  may  therefore 
easily  be  conceived  to  differ  in  any  imaginable  degree  in  different  media,  and  there  is,  at  least,  no  absurdity  in  sup- 
posing the  ether  so  constituted  as  to  admit  of  comparatively  very  feeble  lateral  propagation.  Now,  thirdly,  in  point 
of  fact,  light  does  spread  itself  in  a  certain  small  degree  into  "the  shadows  of  bodies,  out  of  its  strict  rectilinear 
course,  giving  rise  to  the  phenomena  of  inflexion  or  diffraction,  of  which  more  presently,  and  which  are  com- 
pletely accountable  for  on  the  undulatory  doctrine,  and  form,  in  fact,  its  strongest  points.  For  further  informa- 
tion on  this  confessedly  abstruse  subject,  the  reader  must  consult  our  article  on  SOUND,  and  the  works  cited  at 
the  end  of  this  Essay.  It  is  enough  here  to  show,  that  the  objection  which  has  been  urged  by  Newton  and  his 
followers  with  such  force  against  the  doctrine  of  undulations,  is  really  not  conclusive  against  it,  but  founded 
rather  on  inadequate  conceptions  of  the  nature  of  elastic  fluids,  and  the  laws  of  their  undulations. 

Although  any  kind  of  impulse,  or  motions  regulated  by  any  law,  may  be  transferred  from  molecule  to 
molecule  in  an  elastic  medium,  yet  in  the  theory  of  light  it  is  supposed  that  only  such  primary  impulses  as  recur 
according  to  regular  periodical  laws,  at  equal  intervals  of  time,  and  repeated  many  times  in  succession,  can 
affect  our  organs  with  the  sensation  of  light.  To  put  in  motion  the  molecules  of  the  nerves  of  our  retina  with 
sufficient  efficacy,  it  is  necessary  that  the  almost  infinitely  minute  impulse  of  the  adjacent  ethereal  molecules 
should  be  olten  and  regularly  repeated,  so  as  to  multiply,  and,  as  it  were,  concentrate  their  effect.  Thus,  as  a 
great  pendulum  may  be  set  in  swing  by  a  very  minute  force  often  applied  at  intervals  exactly  equal  to  its  time 


Pan  III. 


LIGHT.  451 

of  oscillation,  or  as  one  elastic  solid  body  can  be  set  in  vibration  by  the  vibration  of  another  at  a  distance,  Parl  1J'- 
""~v~'  propagated  through  the  air,  if  in  exact  unison,  even  so  may  we  conceive  the  gross  fibres  of  the  nerves  of  the  >"^~V~~' 
retina  to  be  thrown  into  motion  by  the  continual  repetition  of  the  ethereal  pulses ;  and  such  only  will  be  thus 
agitated,  as  from  their  size,  shape,  or  elasticity  are  susceptible  of  vibrating  in  times  exactly  equal  to  those  at 
which  the  impulses  are  repeated.  Thus  it  is  easy  to  conceive  how  the  limits  of  visible  colour  may  be  established ; 
for  if  there  be  no  nervous  fibres  in  unison  with  vibrations  more  or  less  frequent  than  certain  limits,  such  vibra- 
tions, though  they  reach  the  retina,  will  produce  no  sensation.  Thus,  too,  a  single  impulse,  or  an  irregularly 
repeated  one,  produces  no  light ;  and  thus  also  may  the  vibrations  excited  in  the  retina  continue  a  sensible 
time  after  the  exciting  cause  has  ceased,  prolonging  the  sensation  of  light  (especially  of  a  vivid  one)  for  an 
instant  in  the  eye  in  the  manner  described,  (Art.  543.)  We  may  thus  conceive  the  possibility  of  other  animals, 
such  as  insects,  incapable  of  being  affected  with  any  of  our  colours,  and  receiving  their  whole  stock  of  luminous 
impressions  from  a  class  of  vibrations  altogether  beyond  our  limits,  as  Dr.  Wollaston  has  ingeniously  imagined 
(we  may  almost  say  proved)  to  be  the  case  with  their  perceptions  of  sound. 

The  law  of  motion  of  every  particle  of  the  ether  is  regulated  by  that  of  the  molecule  of  the  luminary  from      568. 
which  it  takes  its  origin ;    and  will  be  regular  or  irregular,  periodical  or  not,  according  as  that  of  the  original  Motion  of  i 
molecule  is  so  or  otherwise.     But  it  is  only  with  motions  which  may  be  regarded  as  infinitely  small   that  we  i^^1"^ 
are    concerned   in  this    theory.     The   displacement  of  each  particle,  either  of  the  ether  or  of  the   luminary,  is  mo]ecuie 
supposed  to  be  so  minute  as  not  to  detach  it  from,  or  change  its  order  of  situation  among  the  neighbouring 
ones.     Now  when  we  consider  only  such   infinitesimal  displacements   from   the  position  of  equilibrium,  it  is 
evident,  that  the  tension  arising  from  them,  or  the  force  by  which   the  displaced  molecule  is  urged,  must  be 
proportional  in  quantity  to  its  distance  from  its  point  of  rest,  and  must  tend  directly  to  that  point,  provided  we 
suppose  the  medium  equally  elastic  in  all  directions.     Hence,  by  the  laws  of  Dynamics,  its  trajectory  must  be  an 
ellipse  described  in  one  plane  about  the  point  of  equilibrium  as  its  centre ;   or,  if  one  of  the  axes  of  the  ellipse 
vanish,  a  straight  line  having  that  point  in  its  middle,  in  which  it  oscillates  to  and  fro,  performing  all  its  excur- 
sions in  the  latter  case,  or  its  revolutions  in  the  former,  whether  great  or  small,  in  equal  times,  and  following  the 
law  of  a  vibrating  pendulum.     We  will,  for  the  present,  consider  the  case  of  rectilinear  vibrations  as  the  most 
simple,  and  show  hereafter  how  the  more  general  one  may  be  reduced  to  it. 

Proposition.  To  define  the  motion  of  a  vibrating  molecule  of  a  luminary,  supposing  its  excursions  to  and  fro       559. 
to  be  performed  in  straight  lines.  Laws  of 

Putting  x  for  its  distance  from  its  point  of  rest,  t  for  the   time  elapsed   since  a  given  epoch,  and  v  for  its  rectilinear 
velocity,  and  E  for  the  absolute  elastic  force,  the  force  urging  the  molecule  to  its  point  of  equilibrium  will  be 
E  .  JT,  and  will  tend  to  diminish  x;  hence  (supposing  gravity  to  be  represented  by  32 £  feet)  we  must   have 


— — —  =  -     -T-TT-  =  E  x,  and  therefore  -  — •  =  —  2  E  xdx,  or,  integrating,  —      -  or  c2    =  E 

dtat  d  t  d  t* 

(a*  —  i2)  where   a  is  the    greatest   distance  of  excursion,  or  the   semiamplitude   of  the  vibration.      Hence, 


1 

—  arc  .  cos  -  ,  that  is 

a 


x  =  a  .  cos  {  </E  .  (t  +  C)  }  ;         v  =  a  .  -/E   .  sin  {  */E  (t  -f  C)  } 

Such  are  the  velocity  and  distance  from  the  middle  point  of  its  vibration  of  the  molecule  at  any  instant.  If 
we  call  T  the  whole  period  in  which  the  molecule  has  performed  one  complete  evolution,  consisting  of  a 
complete  excursion  to  and  fro  on  both  sides  of  its  point  of  equilibrium,  we  shall  have  at  the  commencement  of 

the  motion  when  v  =  0,  or  x  =  a,  a  .  cos  {  vE  .  (t  -f-  C)  }  =  a,  or  (t  -f-  C)  '/E  =  0  ;  and  when  one  quarter 
of  a  period  has  been  performed,  or  the  molecule  has  arrived  at  its  greatest  distance  —  a  on  the  opposite  side  of 
the  centre  -  a  =  a  .  cos  {  V'E  (t  -f  i  T  4-  C)  }  .  or  VE  .  (t  +  C  -f  \  T)  =  w,  putting  v  for  the  semicircum- 
ference  of  a  circle  whose  diameter  is  1  .  Hence  we  get  by  subtraction 


.—      •  • 

V  E 
Hence  we  may  eliminate  E,  and  introduce  T  instead  of  it,  which  will  give  the  equations  ^~E  = 

t-i-C  . —  t  +  C 

x  =  a  .  cos  2  TT  .  — i- —  ;          v  =  a  v  E  sm   2  TT  .  —± — ; 


—  , 


which  equations  express  the  laws  required,  and  which  if  the  time  t  be  supposed  to  commence  at  the  moment 
when  v  =  0,  or  when  the  molecule  is  at  the  extremity  of  one  of  its  excursions,  become  simply 


x  =  a  .  cos  2  tr .  -—-  •  v  =  a  •/  E  sin  2  v .  — • 


3N2 


452 


LIGHT. 


Light. 

— -V* 
570. 


571. 
Laws  of 
rectilinear 
vibrations 
of  an 
ethereal 
molecule. 


572. 


573. 
Waves  of 
light 

denned. 


Carol.  Hence  the  excursions  of  the  molecule  to  and  fro  will  consist  of  four  principal  phases,  in  each  of  which 
'  its  motion  is  similar,  but  in  contrary  directions,  or  on  contrary  sides  of  the  centre.  In  the  first  phase  the 
molecule  is  to  the  right  of  the  centre  of  motion,  and  is  approaching  the  centre,  or  moving  from  right  to  left. 
In  the  second,  it  is  to  the  left  of  the  middle  point,  and  moving  from  it,  or  still  from  right  to  left.  These  two 
phases  we  shall  term  the  positive  phases.  In  the  third  phase  the  molecule  lies  on  the  left  side,  and  its  motion  is 
towards  the  centre,  and  from  left  to  right.  In  the  fourth,  it  is  to  the  right  again,  receding  from  the  centre,  and 
moving  still  from  left  to  right.  These  we  shall  term  the  negative  phases  of  its  vibration. 

Proposition.  To  define  the  rectilinear  vibrations  of  any  molecule  of  the  ether,  propagated  from  a  luminous 
particle  vibrating  as  in  the  last  proposition. 

In  the  propagation  of  motions  through  elastic,  uniform  media,  the  same  or  a  similar  motion  to  that  of  any 
one  molecule  is  communicated  to  every  other  in  succession ;  but  this  communication  occupies  time,  and  the 
motion  of  a  molecule  at  a  distance  from  the  origin  of  the  vibrations  does  not  commence  till  after  the  lapse  of  an 
interval  of  time  proportional  to  that  distance,  being  the  time  in  which  the  propagated  impulse,  whether  of  sound 
or  light,  &c.  runs  over  that  distance  with  a  certain  uniform  velocity  due  to  the  intrinsic  elasticity  of  the  medium, 
and  which  in  the  case  of  light  is  about  200,000  miles  per  second  ;  in  that  of  sound  about  1100  feet.  And  when 
the  vibration  of  the  original  source  of  motion  has  ceased,  that  of  the  ethereal  molecule  does  not  cease  on  the 
instant,  but  continues  for  a  time  equal  to  that  which  elapsed  before  its  commencement.  Hence,  if  we  call  V  the 

velocity  of  light,  and  D  the  distance  of  the  molecule  from  the  luminous  point,  will  be  the  interval  between 

the  commencement  of  the  motion  of  the  latter  and  of  the  former ;   hence  —  t  being  the  time  elapsed  at  any 
instant  since  the  commencement  of  the  first  positive  phase  of  the  vibration  of  the  luminous  point,  t  — 

will  be  the  corresponding  time  in  the  case  of  the  ethereal  molecule.     Thus  we  have,  for  the  equations  of  the 
motions  of  the  former, 

a  =  a  .  cos  2  it  .  -=-  ;  c  =  6  .  sin  2  ir    -  — ;  where  b  =  a   v'  E 

and  in  that  of  the  latter 


Part  HI. 


574. 

Undula- 
tions or 
pulses. 
575. 

Different 
colours 
have  dif- 
ferent 
lengths  of 
their  undu 
lations. 


X  =  a  .  cos  2  IT  . 


v  •=.  ft  .  sin  2 


where  8  —  a 


a  being  the  semiamplitude  of  the  vibration,  or  the  extent  of  the  excursion  of  the  ethereal  molecule  from  its 
point  of  rest. 

Carol.  Hence  it  is  evident  that  the  actual  velocity  of  the  molecules  of  ether  may  be  less  in  any  proportion 
than  that  of  light ;  for  the  maximum  value  of  v  depends  for  its  numerical  magnitude  solely  on  a,  or  on  the 
amplitude  of  excursion,  and  on  E,  and  not  at  all  on  V  the  velocity  of  propagation  of  the  wave. 

Carol.  2.  If  we  suppose  the  luminous  molecule  to  have  made,  from  the  commencement  of  its  motion,  any 
number  of  vibrations  and  parts  of  a  vibration  in  the  time  t ;  then  if  we  consider  an  ethereal  molecule  at  a 
distance  V  .  t  from  it  in  any  direction,  (i.  e.  situated  in  a  spherical  surface  whose  radius  is  V  .  t,)  this 
molecule  will  be  just  beginning  to  be  put  in  motion.  If  we  suppose  another  spherical  surface  concentric  with 
the  former,  but  having  its  radius  less  than  the  former  by  V  .  T,  which  in  future  we  shall  call  \,  every  particle 
situated  in  this  surface  will  have  just  completed  one  vibration,  and  be  commencing  its  second,  and  so  on. 
The  interval  between  these  surfaces  will  comprehend,  arranged  in  spherical,  concentric  shells,  molecules  in 
every  phase  of  their  vibrations, — those  in  each  shell  being  in  the  same  phase.  This  assemblage  of  molecules 
is  termed  a  wave,  and  as  the  impulse  continues  to  be  propagated  forwards  it  is  evident  that  the  wave  will 
continue  to  increase  in  radius,  and  will  comprehend  in  succession  all  the  molecules  of  the  medium  to 
infinity. 

Definition.  The  interval  between  the  internal  and  external  surface  of  a  luminous  wave  is  called  an  undulation, 
or  a  pulse,  and  its  length  is  evidently  =  V .  T  —  X,  or  the  space  run  over  by  light  in  the  time  T  of  one  complete 
period,  or  vibration  of  the  luminous  molecule.  It  is  therefore  proportional  to  that  time. 

Hence  the  lengths  of  the  undulations  of  differently  coloured  rays  differ  inter  se.  For,  by  Postulate  6,  the 
number  of  vibrations  made  in  any  given  time  by  the  ethereal  particles  determines  the  colour.  Now  the  more 
numerous  the  vibrations  are,  data  tempore,  the  shorter  their  duration ;  hence  T,  which  represents  this  duration, 
is  less  ;  and  therefore  X,  or  the  length  of  the  undulation  less  for  the  violet  than  for  the  red  rays.  From 
experiments  to  be  presently  described,  it  has  been  found,  that  the  lengths  of  the  undulations  in  air,  or 
the  values  of  X  for  the  different  rays,  as  also  the  number  of  times  they  are  repeated  in  one  second,  are  as 
in  the  following  table  : 


LIGHT. 


453 


Light. 


Colours. 

Length  of  an  undulation 
in  part;  of  an  inch  in 
air  X  =. 

Number  of  such  undu- 
lations in  an  inch  or  — 

Number  of  undulations  per  second. 

Extreme   

0-0000266 

37640 

458,000000,000000 

Red  

0-0000256 

39180 

477,000000,000000 

Intermediate  

0-0000246 

40720 

495,000000,000000 

Orancre.  . 

0-0000240 

41610 

506,000000,000000 

Intermediate  

00000235 

42510 

517,000000,000000 

Yellow  

0-0000227 

44000 

535,000000,000000 

Intermediate  

0-0000219 

45600 

555,000000,000000 

Green   

0-0000211 

47460 

577,000000,000000 

Intermediate  

0-0000203 

49320 

600,000000,000000 

Blue  

0-0000196 

51110 

622,000000,000000 

Intermediate  . 

0-00001&9 

52910 

644,000000,000000 

Indigo  .  . 

0-0000185 

54070 

658,000000,000000 

Intermediate  

0-0000181 

55240 

672,000000,000000 

Violet   

0-0000174 

57490 

699,000000,000000 

Extreme   

0-0000167 

59750 

727,000000,000000 

Taking   the  velocity   of  light  at 
192000  miles  per  second.    . 

Part  IH 


From  this  table  we  see,  that  the  sensibility  of  the  eye  is  confined  within  much  narrower  limits  than  that 
of  the  ear,  the  ratio  of  the  extreme  vibrations  being  nearly  1'58  :  1,  and  therefore  less  than  an  octave,  and  about 
equal  to  a  minor  sixth.  That  man  should  be  able  to  measure,  with  certainty,  such  minute  portions  of  space  and 
time,  is  not  a  little  wonderful ;  for  it  may  be  observed,  whatever  theory  of  light  we  adopt,  these  periods  and  these 
spaces  have  a.  real  existence,  being,  in  fact,  deduced  by  Newton  from  direct  measurements,  and  involving  nothing 
hypothetical  but  the  names  here  given  them. 

The  direction  of  a  ray  in  the  undulatory  system  is  a  line  perpendicular  to  the  surface  of  the  wave  at  any 
point.  When,  therefore,  the  vibration  is  propagated  through  an  uniform  ether,  the  wave  being  bounded  by 
spherical  surfaces,  the  direction  of  the  ray  is  constant,  and  from  the  centre.  Thus  in  this  system  a  ray  of  light 
moves  in  a  right  line  in  an  uniform  medium. 

The  intensity  of  a  ray  is,  of  course,  in  some  certain  determinate  ratio  of  the  impulse  made  on  the  retina  data 
tempore  by  the  ethereal  molecules,  and  therefore  in  some  certain  ratio  of  their  amplitudes  of  excursion,  or  their 
absolute  velocities.  The  principle  of  the  conservation  of  living  forces  requires  that  the  amplitude  of  excursion 
of  a  molecule,  situated  at  any  distance  from  the  vibrating  centre,  should  be  as  the  distance  inversely,  (see 
ACOUSTICS.)  If  then  we  suppose  the  sensation  created  in  the  retina  to  be  as  the  simple  vis  intrtia  of  the  mole- 
cules producing  it,  light  ought  to  decrease  inversely  as  the  distance  ;  if  as  the  vis  viva,  (which  is  as  the  square  of 
the  velocity,)  inversely  as  the  square  of  the  distance.  As  we  know  nothing  of  the  mode  in  which  the  immediate 
sensation  of  light  or  sound  is  produced  in  the  sensorium,  we  have  no  reason  to  prefer  one  of  these  ratios  to  the 
other  a  priori.  But  when  we  consider,  that  in  the  division  of  a  beam  of  light  by  partial  reflexion,  or  by  double 
refraction,  or  otherwise,  there  is  neither  gain  nor  loss  of  light,  (supposing  the  perfect  transparency  and  polish  of 
the  medium  which  operates  the  division)  so  that  the  sum  of  the  intensities  remains  constant,  however  the  absolute 
velocities  of  the  vibrating  molecules  may  change,  either  in  quantity,  or  (as  in  the  case  of  reflexion,  where  they 
must  be  conceived  to  rebound  from  each  other,  mediately  or  immediately)  in  sign,  the  agreement  of  this  law  in 
all  cases  with  that  of  the  conservation  of  the  v's  viva,  and  its  opposition  in  the  other  mentioned  case  to  that  of 
the  uniform  motion  of  the  centre  of  gravity,  (which  would  make  not  the  sum,  but  the  difference  of  the  intensities 
constant,  were  the  simple  ratio  of  their  velocities  assumed  for  their  measure,)  (see  DYNAMICS,)  leaves  us  no 
choice  in  preferring  the  square  of  the  absolute  velocity,  or  of  the  amplitude  of  excursion  of  a  vibrating  molecule, 
for  the  measure  of  the  intensity  of  the  ray  it  propagates  ;  and  thus  the  observed  law  of  the  diminution  of  light 
is  reconciled  to  the  undulatory  doctrine. 

When  the  medium  through  which  the  vibrations  are  transmitted  is  not  uniformly  elastic,  the  waves  will  make 
unequal  progress  in  different  directions,  according  to  the  law  of  elasticity.  In  this  case  the  figure  of  the  wave 
will  not  be  spherical.  If  we  suppose  the  elasticity  to  vary  by  insensible  gradations,  as  when  light  passes  through 
the  atmosphere,  whose  refracting  power  is  variable,  the  figure  of  the  wave  will  be  flattened  towards  that  part 
where  the  elasticity  is  less.  Thus,  in  fig.  126,  if  A  B  be  the  earth's  surface,  C  D,  E  F,  G  H,  &c.  the  atmo- 
spheric strata,  and  S  a  luminous  point,  the  waves  will  be  less  curved  as  they  approach  the  perpendicular  S  B  ;  and 
the  line  S,  1,2,  3,  4,  5,  &c.  drawn  so  as  to  intersect  them  all  at  right  angles,  will  be  a  curve  convex  downwards, 
so  that  a  ray  will  appear  to  be  continually  bent  downwards  towards  the  earth,  as  we  see  really  happens.  Let  us 
now  proceed  to  consider  the  explanation  of  the  phenomena  of  reflexion  and  refraction  on  the  undulatory  system. 
The  perpendicular  reflexion  of  light  may  be  conceived,  by  the  analogy  of  an  elastic  ball  in  motion  impinging 
directly  on  another  at  rest,  and  in  this  way  it  has  been  illustrated  by  Dr.  Young.  If  the  balls  be  equal,  the 
whole  motion  of  the  impinging  ball  will  be  transferred  to  the  other,  no  reflexion  taking  place  ;  and  thus  the 
impulse  may  be  propagated  undirninished  along  a  line  of  balls  as  far  as  we  please.  So  it  is  with  light  moving 
in  a  uniform  medium,  or  passing  from  one  medium  to  another  of  equal  elasticity.  But  if  a  less  ball  impinge  OQ 


576 


577. 

Direction  of 
a  ray. 

578. 

Law  of  in- 
tensity of 
light 


579. 

Form  of  the 
wave. 


580. 

Perpendi- 
cular re- 


454  L  I  G  H  T. 

Light,     a  greater  at  rest,  it  will  be  reflected,  and  with  a  momentum  which  is  greater  in  proportion   to  the  difference  in    Pun  ill. 
v— -V— '  size  of  the  balls.  %— ^^_ 

581.  But  to  render  an  account  of  oblique  reflexion  and  refraction,  and  the  other  phenomena  we  shall  have  to  speak 
ncip  es.    of(  jt  w|j|  jje  necessary  to  ]ay  down  the  following  principles,  which  are  either  self-evident  or  follow  immediately 

from  the  elementary  principles  of  dynamics. 

582.  1.  When  any  number  of  very  minute  impulses  is  communicated   at  once  to  the  particles  of  any  medium,  or 
Superpo-     of  any  mechanical  system  under  the  influence  of  any  forces,  the  motion  of  each  particle  at  any  instant  will  be  the 
small           Sum  °^  a"  t'le  mot'ons  which  it  would  have  at  that  instant,  had  each  of  the  impulses  been  communicated  to  the 
motions.       system  alone,  (the  word  sum  being  understood  in  its  algebraical  sense.) 

583.  2.  Every  vibrating  molecule  in  an  elastic  medium,  whether  vibrating  by  an  original  impulse,  or  in  consequence 
Principle  of  of  an  impulse  propagated  to  it  from  others,  may  be   regarded  as  a  centre  of  vibration  from  which  a  system  of 
secondary    secondary  waves  emanates  in  all-directions,  according  to  the  laws  of  the  propagation  of  waves  in  the  medium. 

Proposition.  In  the  reflexion  of  light  on  the  undulatory  doctrine,  the  angle  of  incidence  is  equal  to  that  of 
reflexion. 

584.  Let  A  B  be  a  plane   surface  separating  the  two  media,  and  S   the  luminous   point  propagating  a  series  of 
Law  of  re-    spherical  waves,  of  which  let  A  a  be  one.     So  soon  as  this  reaches  the  surface  at  A,  a  partial  reflexion  will  take 

"°°  at  a  place  ;  and  regarding  the  point  A  as  a  new  centre  of  vibration,  spherical  waves  will  begin  to  be  propagated 
from  it  as  a  centre,  one  of  which  proceeds  forwards  into  the  reflecting  medium,  with  a  velocity  greater  or  less 
than  that  of  the  incident  wave,  as  the  case  may  be  ;  the  other  backwards  into  the  medium  of  incidence,  with  a 
velocity  equal  to  that  of  the  incident  wave.  It  is  only  with  the  latter  we  are  at  present  concerned.  Conceive 
now  the  wave  A  a  to  move  forward  into  the  position  B  6  ;  then  in  the  time  that  it  has  run  over  the  space  P  B,  the 
wave  propagated  from  A  will  have  run  back  over  a  distance  A  d  =  P  B,  and  the  hemisphere  whose  radius  is  A  d 
will  represent  this  wave.  Between  A  and  B  take  any  point  X,  and  describe  the  hemispheric  surface  X  c.  Then 
regarding  X  as  a  centre  of  vibration,  its  vibrations  will  not  commence  till  the  wave  has  reached  it.  It  will,  there- 
fore, begin  to  vibrate  later  than  A,  by  the  whole  time  the  wave  A  a  takes  to  run  over  P  Q  ;  but  when  once  set 
in  vibration,  it  propagates  backwards  a  spherical  wave  with  the  same  velocity,  so  that  when  the  original  wave 
has  advanced  into  the  situation  B  b,  the  wave  from  X  will  have  expanded  into  a  hemisphere,  whose  radius  X  c 
is  equal  toPB,  —  PQ,  or  A  B.  Now  this  being  true  of  every  point  X,  if  we  conceive  a  surface  touching  all  these 
hemispheres  in  d,  c,  B,  this  surface  will  mark  the  points  at  which  the  reflected  impulse  has  just  arrived,  and 
which  just  begins  to  move  when  the  original  wave  has  reached  B,  and  will,  therefore,  be  the  surface  of  the 
reflected  wave.  Conceive  now  the  spherical  surface  6  B  prolonged  below  the  plane  A  B,  as  represented  by 
the  dotted  line  D  C  B,  and  the  same  of  the  spheres  about  A  and  X.  Then  the  spherical  surfaces  D  C  B  and 
C  c  being  both  perpendicular  to  S  X  C,  must  touch  each  other  in  C,  hence  the  surface  touching  all  the  hemi- 
spheres about  A,  X,  &c.  as  centres,  below  A  B  is  a  segment  of  a  sphere  having  S  for  a  centre,  and  therefore 
the  surface  B  erf  or  the  reflected  wave  is  a  segment  of  a  sphere  having  its  centre  at  s  as  much  below  the  line 
A  B  as  S  is  above  it. 

Now  to  an  eye  placed  at  X,  the  luminous  point  S  will  appear  in  the  direction  S  X  perpendicular  to  the  incident 
wave,  and  the  eye  placed  in  c  will  perceive  the  reflected  image  of  S  at  *  in  the  direction  cs,  perpendicular  to 
the  reflected  wave  ;  but  cs  passes  through  X,  because  the  spheres  cC  and  B  b  touch  at  c.  Therefore  the  ray  by 
which  *  is  seen  at  c  passes  through  X.  But  the  surfaces  B  D,  B  d  being  similar  and  equal,  the  angle  B  X  c  = 
B  X  C  =  A  X  S,  that  is,  the  angle  of  incidence  is  equal  to  that  of  reflexion.  Q.  E.  D. 

585.  Cor.     If  the  reflecting  surface  be  not  a  plane,  the  reflected  wave  will  not  be  spherical ;  its  form  is,  however, 
Reflexion  at  easily  determined  as  follows :  Suppose  the  direct  wave  to  have  assumed  the  position  B  6.     Take  any  point  X  in 
curved  sur-   the  reflecting  surface,  and  describe  the  sphere  X  Q,  and  with  the  centre  X  and  radius  =  B  Q,  describe  another 
taces.           sphere.     Do  this  for  every  point  in  the  surface  A  B,  and  the   surface  which  is  a  common   tangent  (as  B  c  d)  to 

ig.  128.  ajj  these  spheres,  is  the  surface  of  the  reflected  wave,  because  it  marks  the  farthest  limit  to  which  the  reflected 
impulse  has  reached  in  all  directions  at  the  instant  when  the  direct  impulse  has  reached  B.  Now  take  Y 
infinitely  near  to  X,  and,  making  the  same  construction  at  Y,  let  c,  e.  be  the  points  in  the  reflected  wave  to  which 
X  c  and  Y  «  are  respectively  perpendicular.  Draw  X  r  perpendicular  to  Y  e,  and  X  q  to  SYq,  then,  since  Y  e  = 
SB  -  SY,  andXe=  SB  -  SX,  we  have  Y  e  -  Xc,  or  Y  r  =  S  X  -  S  Y  =  Y  q,  and  X  Y  being  common  to 
the  right  angled  triangles  X  Y  r,  X  Y  q,  the  angle  r  Y  X  must  be  equal  to  X  Y  q  or  to  S  Y  A,  so  that  the  same 
law  of  reflexion  holds  good  in  curve  as  in  plane  surfaces. 

586.  Proposition.     To  demonstrate  the  law  of  refraction  in  the  undulatory  system. 

Let  S,  fig.  129,  be  a  luminous  point,  and  let  any  wave  propagated  from  it  reacli  in  succession  the  points  Y, 
^'  ^  °^  any  curve  surface  Y  X  B  of  a  refracting  medium,  whereof  X  and  Y  are  supposed  infinitely  near  each 
other.  As  the  wave  strikes  Y,  X,  B,  each  of  these  points  will  become  centres  of  undulation,  which  will  be 
propagated  in  the  refracting  medium  with  a  velocity  different  from  that  of  light  in  the  medium  of  incidence,  by 
reason  of  their  different  elasticities,  (Postulate  3.)  Let  V  :  v  '.  \  velocity  in  the  first  medium  to  that  in  the 

second,  (a  constant  ratio  by  hypothesis,)  and,  describing  the  sphere  B  Q  R,  take  X  c  =  -^-  .  Q  X  and  Y  e  = 

-=r-  .  V  R,  then  will   X  c  and  Ye  represent  the  spaces  run  over  by  the   refracted  secondary  waves  propagated 

from  X  and  Y  respectively,  when  the  direct  wave  has  reached  B.  Hence,  if  about  X  and  Y  as  centres,  and  with 
these  radii  we  describe  spheres,  and  suppose  e,  c  to  be  points  in  the  curve  surface  which  is  a  tangent  to  all  such 
spheres,  it  is  clear  that  X  cand  Ye  will  be  perpendicular  to  this  surface,  that  is,  to  the  surface  of  the  refracted  primary 
wave  ;  hence,  X  c  and  Y  e  will  be  the  directions  of  the  refracted  rays  at  X  and  Y.  Draw  X  q,  Xr  perpendiculai 


LIGHT.  455 

respectively  to  Y  R  and  Y  e,  then  will  Y<?=:SX-SYandYr  = 


(YR-XQ)  =  ^-  {(SR-SY)-  (SQ   -  S  X)  }  =     -  .  (S  X  -  S  Y)  =  ~  .  Y  q.     Hence  we  have  Y?  : 

Yr  ;:  V  :  v.     But  since  S  X,  SY  are  direct  rays,  and  X  o,  Y  e  the  corresponding  refracted  ones,  therefore 
S  X  Y  is  the  complement  of  the  angle  of  incidence  of  S  X,  and,  consequently,  Y  X  q  is  equal  to  the  angle  of 
incidence  itself,  and  X  Yr  will  be  the  complement  of  the  angle  of  refraction,  and    therefore   Y  X  r  (  =  90°  - 
X  Y  r)  =  the  angle  of  refraction  of  S  Y,  or,  (since  the  points  Y,  X  are  infinitely  near  each  other,)  of  S  X,  hence 
we  have 

Y  q  :  X  Y  I  '.  sin  incidence  :  1, 
X  Y  :  Y  r  !  I  1  :  sin  refraction. 

And  compounding  Y  q  :  Y  r  '.  '.  sin  incidence  :  sin  refraction.     But  we  proved  before,  that  Y  q  :  Y  r  in  the 
constant  ratio  of  V  :  v  ;  therefore  the  sine  of  incidence  :  that  of  refraction  in  the  same  constant  ratio.   Q.  E.  D. 

Corollary  I.  In  the  cases  both  of  reflexion  and  refraction,  the  undulation  is  propagated  from  the  luminous  point  587. 
to  any  other  point  in  the  least  possible  time.  For  the  surface  both  of  the  reflected  and  refracted  waves  mark  the 
extreme  limits  to  which  the  impulse  has  been  propagated  by  reflexion  or  refraction  in  a  given  time.  The  undu- 
lation propagated  from  X  (fig.  127)  in  any  other  direction  than  X  c,  as,  for  instance,  X  7,  will  fall  short  of  the  surface 
B  erf,  and  the  point  7  therefore  will  have  been  reached,  and  passed  by  the  reflected  or  refracted  primary  wave  in 
the  situation  ft  7  S,  before  it  can  be  reached  by  the  secondary  undulation  propagated  from  X  in  the  direction  X  7. 

Corollary  2.     This  property  in   the  undulatory  system  corresponds  to   the  principle   of  least  action  in  the      58S. 
corpuscular  doctrine,  and  may  be  thus  stated  generally  :  Law  of 

A  reflected  or  refracted  ray  will  always  pursue  such  a  course  as  would  be  described  in  the  least  possible  time,  «*ifte« 
by  a  point  moving  from  the  point  of  its  departure  to  that  of  its  arrival,  with  the  velocities  corresponding  to  the  prt" 
media  in  which  it  moves,  and  the  direction  of  its  motion. 

It  is  evident  that  this  is  general,  and  applies  to  cases  where  the  medium  is  either  of  variable  elasticity,  or  has      589. 
different  elasticities  in  different  directions;  for  the  ray  is  by  definition  a  perpendicular  to  the  surface  of  the  wave,  Applies 
or  to  a  surface,  the  locus  of  all  the  molecules  in  the  medium,  which  are  just  attained  by  the  undulation,  and  just  g™era"y 
commencing  their  vibration,  so  that  the  reasoning  of  Corel.  1,  applies  equally  to  all  cases. 

The  properties  of  foci  and  Caustics  flow  with  such  elegance  and  simplicity  from  this  doctrine,  that  it  would      590. 
be  unpardonable  not  to  instance  its  application  to  that  part  of  the  theory  of  Optics. 

Definition.     A  focus  is  a  point  at  which  the  same  wave  arrives  at  the  same  instant  from  more  than  one  point  svstcm 
in  a  surface.  Defined. 

It  is  evident,  that  when  this  is  the  case,  the  ethereal  molecules  in  the  focus  will  be  agitated  by  the  united  force 
of  all  the  undulations  which  reach  them  in  the  same  phase  at  the  same  instant,  and  will  be  proportionally  more 
violent  as  the  focus  is  common  to  a  greater  number  of  points,  and  the  light  in  the  focus  will  be  proportionally 
more  intense. 

Proposition.     Required  to  determine  the   nature  of  the  surface  which  shall   refract  all  rays  from  one  point       59] 
rigorously  to  one  focus.     Let  F  (fig.  129)  be   the  focus,  then  will  every  part  of  a  wave  propagated  from  S  and 
refracted  at  the  surface  A  B,  reach  F  at  the  same  instant  ;   therefore  time  of  describing  S  X  with  velocity  V  + 
time  of  describing  X  F  with  velocity  v  is  constant  for  every  point  in  the  surface.     Or, 

S  X          F  X 

-j  --    —  =  constant,  orSX-j-/*.FX=  constant,  /*  being  the  relative  index  of  refraction. 


V 

This  equation  then  defines  the  nature  of  the  curve  sought,  and  it  is  easy  to  perceive  its  identity  with  that 
expressed  by  the  equation  («)  Art.  232,  obtained  from  a  direct  consideration  of  the  law  of  refraction,  but  by  a 
much  more  intricate  process. 

The  intensity  of  the  reflected  or  refracted  ray  cannot  be  computed  generally  in  the  present  very  imperfect  state      593. 
of  our  knowledge  of  the  theory  of  waves.     M.  Poisson,  however,  in  the  case  of  perpendicular  incidence,  and  on  Intensi 


the  particular  hypothesis  of  the  luminous  vibrations  being   performed  in   the  direction  of  the  ray  itself,   has  a  ray  re- 
succeeded  in  investigating  the  comparative  intensities  of  the  incident,  reflected,    and  transmitted  rays.     His   ^ji(.J"  ' 
results  are  as  follows  :    Taking  p.,  /*'  for  the  absolute  refractive  indices  of  the   media,  he  finds  (on  the   supposi-  lar|y 


tion  that  the  intensity  of  light  is  as  the  square  of  the  absolute  velocity  of  the  vibrating  molecules)  : 

Intensity  of  reflected  ray  :  that  of  incident  ;  ;  (/  —  ,u)z  :  (//  +  ^)'J.  Intensity  of  the  intromitted  ray  :  that 
of  the  incident  ;  '.  4/ta  :  (/»  -f-  A1')4-  Intensity  of  the  ray  intromitted  from  a  medium  whose  refractive  index  ==  /i 
into  a  parallel  plate  of  one  whose  refractive  index  =  p',  in  contact  at  its  second  surface  with  a  third  whose  refractive 
index  =  /*'',  reflected  at  their  common  surface,  and  again  emergent  at  the  first  surface  :  intensity  of  the  ray 
originally  incident  on  the  first  surface  ;  ;  16  f  /.'«  (/»''  /.')*  :  O  +  ^')<  .  (/  +  ,u")e.  And>  lastlv>  the  int<;nsity 
of  the  ray  transmitted  through  the  parallel  plate  of  the  second  medium  into  the  third  :  that  of  the  original 
incident  ray  ;  ;  16  /iVJ:  (.»  +  /O2-  (/*'  +  A4")2  which  (in  the  case  where  the  third  medium  is  the  same  as  the 
first,  becomes  16  /i'2  /*'*  :  (/*-)-  ft')*. 

These  results  of  M.  Poisson,  so  far  as  they  have  been  hitherto  satisfactorily  compared  with  experiment,  593. 
manifest  at  least  a  general  accordance,  and  the  undulatory  doctrine  thus  furnishes  a  plausible  explanation  of  the 
connection  of  the  reflecting  power  of  a  medium  with  its  refractive  index,  and  of  the  diminished  reflection  at  the 
common  surfaces  of  media  in  contact.  —  They  have  been  in  great  measure  (it  should  be  observed)  anticipated  by 
Dr.  Young,  in  his  Paper  on  Chromatics,  (Encyclop.  Brit.)  by  reasoning  which  M.  Poisson  terms  indirect,  but 
which,  we  confess,  appears  to  us  by  no  means  to  merit  the  epithet. 


456  LIGHT. 

1/ight.          If  photometrical  experiments  enable  us  to  determine  the  proportion  of  the  reflected  to  the   incident  light,  we    Part  HI. 

— — \/— -^  may  thence  conclude  the  index  <tf  refraction  of  the  reflecting  medium,  and  that  in   cases  where  no   other  mode  v-~"v— •"•" 
594.      wjll  apply.     Thus,  M.  Arago  having  ascertained  that  about  half  the  incident  light  is  reflected  at  a  perpendicular 

Applied  to  /     i  i  i 

irir™\"e  incidence  from  mercury,  we  have  in  this  case  (  —t J  =  £  ;  —  =  5-829  for  the  refractive  index  of  mer- 
cury out  of  air;  and  this  is  perfectly  consonant  to  the  general  tenor  of  optico-chemical  facts,  which  assign  to  the 
heavy  and  especially  to  the  white  metals  (as  indicated  in  (heir  transparent  combinations)  enormous  refractive 
and  dispersive  powers.  This  curious  and  interesting  application  has  not  been  overlooked  by  Dr.  Young  in  the 
Paper  alluded  to. 

595  To  complete  the  theory  of  reflexion  and  refraction  on  the  undulatory  hypothesis,  it  will  be  necessary  to  show 

what  becomes  of  those  oblique  portions  of  the  secondary  waves,  diverging  in  all  directions  from  every  point  of 
the  reflecting  or  refracting  surfaces  (as  X  7,  fig.  127)  which  do  not  conspire  to  form  the  principal  wave.  But 
to  understand  this,  we  must  enter  on  the  doctrine  of  the  interference  of  the  rays  of  light, — a  doctrine  we  owe 
almost  entirely  to  the  ingenuity  of  Dr.  Young,  though  some  of  its  features  may  be  pretty  distinctly  traced  in  the 
writings  of  Hooke,  (the  most  ingenious  man,  perhaps,  of  his  age,)  and  though  Newton  himself  occasionally 
indulged  in  speculations  bearing  a  certain  relation  to  it.  But  the  unpursued  speculations  of  Newton,  and  the 
appercus  of  Hooke,  however  distinct,  must  not  be  put  in  competition,  and,  indeed,  ought  scarcely  to  be 
mentioned  with  the  elegant,  simple,  and  comprehensive  theory  of  Young, — a  theory  which,  if  not  founded  in 
nature,  is  certainly  one  of  the  happiest  fictions  that  the  genius  of  man  has  yet  invented  to  group  together  natural 
phenomena,  as  well  as  the  most  fortunate  in  the  support  it  has  unexpectedly  received  from  whole  classes  of  new 
phenomena,  which  at  their  first  discovery  seemed  in  irreconcileable  opposition  to  it.  It  is,  in  fact,  in  all  its 
applications  and  details  one  succession  of  felicities,  insomuch  that  we  may  almost  be  induced  to  say,  if  it  be  not 
true,  it  deserves  to  be  so.  The  limits  of  this  Essay,  we  fear,  will  hardly  allow  us  to  do  it  justice. 


§  III.     Of  the  Interference  of  the  Rays  of  Light. 

596.  The  principle  on  which  this  part  of  the  theory  of  Light  depends,  is  a  consequence  of  that  of  the  "  Superposition 
General  of  small  motions"  laid  down  in  Art.  583.  If  two  waves  arrive  at  once  at  the  same  molecule  of  the  ether,  that 
principles  of  molecule  will  receive  at  once  both  the  motions  it  would  have  had  in  virtue  of  each  separately,  and  its  resultant 
inference  motjon  w;][)  therefore,  be  the  diagonal  of  a  parallelogram  whose  sides  are  the  separate  ones.  If,  therefore,  the 
two  component  motions  agree  in  direction  or  very  nearly  so,  the  resultant  will  be  very  nearly  equal  to  their  sum, 
and  in  the  same  direction.  If  they  very  nearly  oppose  each  other,  then  to  their  difference.  Suppose,  now,  two 
vibratory  motions  consisting  of  a  series  of  successive  undulations  in  an  elastic  medium,  all  similar  and  equal  to 
each  other,  and  indefinitely  repeated,  to  arrive  at  the  same  point  from  the  same  original  centre  of  vibration,  but 
by  different  routes  (owing  to  the  interposition  of  obstacles  or  other  causes)  exactly,  or  very  nearly  in  the  same 
final  direction  ;  and  suppose,  also,  that  owing  either  to  a  difference  in  the  lengths  of  the  routes,  or  to  a  differ- 
ence in  the  velocities  with  which  they  are  traversed,  the  time  occupied  by  a  wave  in  arriving  by  the  first  route 
(A)  is  less  than  that  of  its  arriving  by  the  other  (B).  It  is  clear,  then,  that  any  ethereal  molecule  placed  in  any 
point  common  to  the  two  routes  A,  B,  will  begin  to  vibrate  in  virtue  of  the  undulations  propagated  along  A, 
before  the  moment  when  the  first  wave  propagated  along  B  reached  it.  Up,  then,  to  this  moment  its  motions 
will  be  the  same  as  if  the  waves  along  B  had  no  existence.  But  after  this  moment,  its  motions  will  be  very 
nearly  the  sum  or  difference  of  the  motions  it  would  have  separately  in  virtue  of  the  two  undulations 
each  subsisting  alone,  and  the  more  nearly,  the  more  nearly  the  two  routes  of  arrival  agree  in  their  final 
direction. 

597  Now  it  may  happen,  that  the  difference  of  the  lengthr,  of  the  routes  or   the  difference  of  velocities  is  such,  that 

Case  of  the  waves  propagated  along  B  shall  reach  the  intersection  exactly  one-half  an  undulation  behind  the  others,  i.  e. 
complete  later  by  exactly  half  the  time  of  a  wave  running  over  a  space  equal  to  a  complete  undulation.  In  that  case, 
discordance  jne  mo]ecuie  which  in  virtue  of  the  vibrations  propagated  along  A  would  (at  any  future  instant)  be  in  one 
phase  of  its  excursions  from  its  point  of  rest,  would,  in  virtue  of  those  propagated  along  B,  if  subsisting 
alone,  be  at  the  same  instant  in  exactly  the  opposite  phase,  i.  e.  moving  with  equal  velocity  in  the  contrary 
direction.  (See  Art.  570.)  Hence,  when  both  systems  of  vibration  coexist  the  motions  will  constantly  destroy 
each  other,  and  the  molecule  will  remain  at  rest.  The  same  will  hold  good  if  the  difference  of  routes  or 
velocities  be  such,  that  the  vibrations  propagated  along  B  shall  reach  the  intersection  of  the  routes  exactly 
*,  ^  ^  &c.  of  a  complete  period  of  undulation  after  those  propagated  along  A;  for  the  similar  phases  of  vibra- 
tion recurring  periodically,  and  being  (by  hypothesis)  continually  repeated  for  an  indefinite  time,  it  is  no  matter 
whether  the  first  vibration  propagated  along  Bbe  superimposed  on,  or  interfere  with  (as  it  is  called)  the  first,  or 
any  subsequent  one  propagated  along  A,  provided  the  difference  of  their  phases  be  the  same. 

598.  On  the  other  hand  it   may  happen,   that  the  waves  propagated  along  B  do  not  reach  the  intersection   till 

Case  of        exactly  one,  two,  or  more  whole  periods  after  the  corresponding  waves  propagated  along  A.     In  this  case,  the 
c-miplete       molecule  at  the  intersection  will,  at  any  instant  subsequent  to  the  time  of  arrival   of  the  first  wave  along  B, 
accordance,  be  agitated  at  once  by  both  vibrations  in  the  same  phase,  and  therefore  the  velocity  and  amplitude  of  its  excur- 
sions will,  instead  of.being  destroyed,  be  doubled. 


LIGHT.  457 

t.ijfht.          Lastly,  it  may  happen,  that  the  difference  of  the  times  of  arrival  of  the  corresponding  waves  is  neither  an    Part  III. 
.— v~^-'  exact  even,  or   odd  multiple  of  half  a  complete  period  of  undulation.     In  that  case,  the   molecule  will  vibrate  *— -v— -^ 
with  a  joint  motion,  less  than  double  what  it  would  have  in  virtue  of  either  separately.  599. 

An  apt  illustration  of  the  case  of  interference  here  described,  may  be  had  by  considering  the  analogous  case  in  600. 
the  interference  of  waves  on  the  surface  of  water.  Conceive,  for  instance,  two  equally  broad  canals  A  and  B  to  illustration 
enter  two  canals  at  right  angles  into  the  side  of  a  reservoir,  at  both  whose  apertures,  from  an  origin  at  a  great  dis-  from  waves 
tance,  a  wave  arrives  at  the  same  instant,  and  runs  along  the  two  canals  with  equal,  uniform  velocities.  Let  their  Propagate 
sides  be  perfectly  smooth,  and  their  breadths  everywhere  equal,  but  let  them  be  led,  by  a  gentle  curvature,  to  meet 
in  a  point  at  some  distance,  and,  the  curvature  of  B  being  supposed  somewhat  greater  than  that  of  A,  let  the 
distance  from  their  intersection  to  the  reservoir,  measured  along  B,  be  greater  than  along  A.  It  is  obvious, 
that  (if  we  consider  only  a  single  wave)  the  portion  of  it  propagated  along  A  will  reach  the  intersection  first, 
and  after  it  that  propagated  along  B,  so  that  the  water  at  that  point  will  be  agitated  by  two  waves  in  succession. 
But,  let  the  original  cause  of  undulation  be  continually  repeated  so  as  to  produce  an  indefinite  series  of  equal 
and  similar  waves.  Then,  if  the  difference  of  lengths  of  the  two  canals  be  just  equal  to  half  the  interval  between 
the  summits  of  two  consecutive  waves,  it  is  evident  that  when  the  summit  of  any  wave  propagated  along  A  has 
reached  the  intersection,  the  depression  between  two  consecutive  summits  (viz.  that  corresponding  to  the  wave 
propagated  along  A,  and  that  of  the  wave  immediately  preceding  it)  will  arrive  at  the  intersection  by  the  course 
B.  Thus,  in  virtue  of  the  wave  along  A  the  water  will  be  raised  as  much  above  its  natural  level,  as  it  will  be 
depressed  below  it  by  that  along  B.  Its  level  will,  therefore,  be  unchanged. — Now  as  the  wave  propagated 
along  A  passes  the  intersection,  it  subsides,  from  its  maximum,  by  precisely  the  same  gradations  as  that  along 
B,  passing  it  with  equal  velocity,  rises,  from  its  minimum,  so  that  the  level  will  be  preserved  at  the  point  of 
intersection,  undisturbed  so  long  as  the  original  cause  of  undulation  continues  to  act  regularly.  So  soon  as  it 
ceases,  however,  the  last  half  wave  which  runs  along  B  will  have  no  corresponding  portion  of  a  wave  along 
A  to  interfere  with,  and  will,  therefore,  create  a  single  fluctuation  at  the  point  of  concourse. 

In  the  theory  of  the  interferences  of  light  we  may  disregard  these  commencing  and  terminal,  uncompensated      601. 
undulations,  and  parts  of  undulations,  as  being  so  few  in  number  as  to   excite  no  impression  on  the  retina,  and  Initial  and 
consider  the  interfering  rays  as  of  indefinite  duration,  or  as  destitute  of  either  beginning  or  end.  terminal  vi- 

According  to  the  foregoing  reasoning  then  it  appears,  that  if  two  rays  having  a  common  origin,  i.  e.  forming  brj 
parts  of  one  and  the  same  system  of  luminous  waves  proceeding  from  a  common  centre,  be  conducted  by  different    °gno 
routes  to  one  point  which  we  will  suppose  to  be  situated  on  a  white  screen,  or  on  the  retina  of  the  eye,  they  Mutual  an- 
will  there  produce  a  bright  point,  or  the  sensation  of  light,  if  their  difference  of  routes  be  an  even  multiple  of  the  nihilation  of 
length  of  half  an  undulation  and  a  dark  one  ;    or  the  sense  of  darkness,  if  an  odd  multiple  of  it;  and  if  inter-  tw<>  rajs  of 
mediate,  then  a  feebler  or  a  stronger  sense  of  light,  as  the  difference  of  routes  approximates  to  one  or  the  other  of  ''S*1.' in  °P" 
these  limits.     That  two  lights  should  in  any  case  annihilate  each  other,  and  produce  darkness,  appears  a  strange  PJ 
paradox,  yet  experiment  confirms  it ;  and  the  fact  was  observed,  and  broadly  stated  by  Grimaldi  long  before  any 
plausible  reason  could  be  given  of  it. 

Having  thus   obtained    a  general  idea  of  the  nature  of  interferences,  let  us  now  endeavour  to  subject  their      603. 
effects  to  a  more  strict  calculation.     To  this  end  it  will  be  necessary  to  fix  with  precision   the  'sense  of  some 
words  hitherto  used  rather  loosely. 

Definition.     The  phase  of  an  undulation  affecting  any  given  molecule  of  ether  at  any  instant  of  time,   is      604. 
numerically  expressed  by  an  arc  of  a  circle  to  radius  unity,  increasing  proportionally  to  the  time— commencing  Definitions. 
at  0  when  the  molecule  is  at  rest  at  its  greatest  positive  distance  of  excursion,  and  becoming  equal  to   one  cir-  pltate- 
cumference  when  the  molecule,  after  completing  the  whole  of  a  vibration,  returns  again  to  the   same  state  of 

/          ^    i    Q  \  t  -  \    c\ 

rest  at  the  same  point.     Thus,  in  the  equation  v  =  a  .  V  E     sin  (  2  jr.  — ^ —  j,  2  jr.  — ^~ —  is  the  phase  of 

the  undulation  at  the  instant  t. 

Definition.     The  amplitude  of  vibration  of  a  ray  or  system  of  waves  is  the  coefficient  a,  or  the  maximum       605. 
excursion  from  rest,  of  each  molecule  of  the  ether  in  its  course.  Amplitude 

Carol.     The  intensity  of  a  ray  of  light  is  as  the  square  of  the  amplitude  of  the  vibrations  of  the  waves  of  which  ot  a  ray- 
it  consists. 

Definition.     Similar  rays,  or  systems  of  luminous  waves,  are  such  as  have  the  vibratory  motions  of  the       606. 
ethereal  molecules  which  compose  them  regulated  by  the  same  laws,  and  their  vibrations  performed  in  equal  Sim'la[ra}'s 
times,  and  the  curves  or  straight  lines  they  describe  in  virtue  of  them,  similar  and  similarly  situated  in  space,  so 
that  the  motions  of  any  two  corresponding  molecules  in  each,  shall  at  every  instant  of  lime  be  parallel  to  each 
other. 

Carol.     Similar  rays  have  the  same  colour. 

Definition.     The  origin  of  a  ray,  or  a  system  of  waves,  is  the  vibrating  material  centre  from  which  the  waves       607. 
begin  to  be  propagated,  or  more  generally,  a  fixed  point  in  its  length,  at  which  an  ethereal    molecule,  at  an  Origin  of  a 
assumed  epoch,  was  in  the  phase  0  of  its  undulation.  ray- 

Carol.  Two  systems  of  interfering  waves  having  their  origins  distant  by  an  exact  number  of  undulations,  may       608. 
be  regarded  as  having  a  common  origin. 

Proposition.     To  find  the  origin  of  a  ray,  having  given  the  expression  for  the  velocity  of  one  of  its  vibrating       609. 

molecules.  To  nnd  tne 

origin  of  a 
/ —  /  t  +  C  \  ray. 

Let  a  =  a  .    v  E,   and   let    v  =  a  .  sin  f  2  v  .  — — 1  be    the    expression    given  for   the  velocity 

VOL.  iv.  3  o 


458  LIGHT. 

Light,      of  any  assumed  molecule    (M)  at  the  instant  t.     Let  V  represent  the  velocity  of  light,  and  \  the  length      !'•>»  HI- 
"•"v-"""  of  an   undulation,    and    S  the  distance  run  over  by  light   in   the  time  t.      Then  will  £  =  V  t  and  X,  =  V  T,  v~^—~> 

and  consequently  —  =  • — .     Suppose  U0  to  represent  the  velocity  of  a  vibrating  molecule  at  the  origin  of  the 
T        X 

ray  at  the  instant  t,  then  will  va  =  a  .  sin  (  2  TT  .  —  j  =  a  .  sin  (  2  JT  --  Y     But  the  molecule  M  moves  only 

by  an  impulse  communicated  to  it  from  the  origin,  and  therefore  all  its  motions  are  later  than  those  at  the  origin 
by  a  constant  interval  equal  to  the  time  required  for  light  to  run  over  the  distance  of  M  from  the  origin.     Call 

D  that  distance,  then  -=-  is  the  interval  in  question,  and  t —  is  the  time  elapsed  at  the  instant  t,  since  the 


0^> 


molecule  commenced  its  periodic  motions;  therefore  its  velocity  v  must  =  a  .  sin  I  2  TT  V  I,  and  con- 
sequently C  = —  ,  Or  D  =  -  V  C. 

Hence  we  see  that  the  distance  of  the  molecule  M  from  the  origin  of  the  ray,  is  equal  to  the  space  described  by 
Light,  in  a  time  represented  by  the  arbitrary  constant  C,  and  is  therefore  given  when  C  is  so,  and  vice  versa. 

610.  Carol.     Since  V  T  =  X  the  expression  for  the  velocity  becomes 

»=a.sin2  JT.(  — —  J  ~  a. .  sin  2  v ( land  similarly  x=  a.  cos  2  a-  f — - —  j 

611.  Proposition.     To  determine  the  colour,  origin,  and  intensity  of  a  ray  resulting  from  the  interference  of  two 
Resultant  of  sjmiiar  raySj  differing  in  origin  and  intensity. 

fering'rays        Let  a*  and  a>  be  the  intensities  of  the  rays,  or  a,  a'  their  amplitudes  of  vibration,  and  take  a  =  a  .  */  E, 
investigated  af  —  a'    A/~E7  then,  if  we  put  0  for  the  phase  of  vibration  of  a  molecule  M  at  the  instant  t  which  it  would  be  in, 

k 

in  virtue  of  the  first  system  of  waves  (A),  and  0  -f  k  for  its  phase,  in  virtue  of  the  other  (B),  ^-  .  T  will  repre- 
sent the  time  taken  by  light  to  run  over  a  space  equal  to  the  interval  of  their  origin,  and  the  velocities 
and  distances  from  rest  which  M  would  have,  separately  at  the  instant  t,  in  virtue  of  the  two  rays,  will  be 

v  =  a  .  sin  0  ;  v'  =  «'  .  sin  (0  +  k),  and  x  =  a .  cos  0 ;  x'  =  a' .  cos  (0  -J-  k). 
Therefore,  in  virtue  of  the  resulting  ray,  it  will  have  the  velocity 

„  _j-  a'  —  a  .  sin  6  +  «' .  sin  (9  +  k),  and  x  -j-  xf  =  a .  cos  0  +  a' .  cos  (0  -j-  k). 

Let  the  former  be  put  equal  to  A  .  sin  (0  +  B),  the  possibility  of  which  assumption  will  be  shown  by  our 
being  able  to  determine  A  and  B,  so  as  to  satisfy  this  condition.  Then  we  have 

(a  -f-  a'  .  cos  k)  sin  0  -\-  a  .  sin  k  .  cos  0  =  A  .  cos  B  .  sin  0  -f-  A  .  sin  B  .  cos  e, 

and  equating  like  terms, 

A  .  cos  B  =  a  -f-  a  .  cos  k ;  A  .  sin  B  =  a' .  sin  k, 

whence  we  get,  dividing  one  by  the  other, 


a'  .  sin  k  a.'  ,  sin  A: 


a  .  sn  a.  ,  sn    :          /— 

tan  B  =  —  7—  r     -7  ;  A  =  -—  -  r  =      "  ~  2  "  °  •  cos  *  +  " 

o-j-o'  .  cosfr  sin  B 

and  these  values  being  determined,  A  and  B  are  known,  and,  therefore,  v  -f-  ^  =  A  .  sin  (0  -j-  B).  Similarly, 
if  we  put  x  +  x'  =  A'  .  cos  (9  +  B')  we  obtain  values  of  A'  and  B'  precisely  similar,  writing  only  a  a1  for  a,  a' 

respectively. 

612  Carol.  1.  Hence  we  conclude,  1st.  that  the  resultant  ray  is  similar  to  the  component  ones,  and  has  the  same 

period,  i.  e.  the  same  colour. 

613  Carol.  2.  M.  Fresnel  has  given  the  following  elegant  rule  for  determining  the  amplitude  and  origin  of  the 

theorem.       resultant  ray,  which  follows  immediately  from  the  value  of  A  and  the  equation  sin  B  =  —  .  sin  K  above  found. 

Construct  a  parallelogram,  having  its  adjacent  sides  proportional  to  the  amplitudes  a,  a'  of  the  component  rays, 
and  the  angle  between  them  measured  by  a  circular  arc  to  radius  unity,  eqiuil  to  the  differences  of  their  phases, 
then  will  the  diagonal  of  this  parallelogram  represent  on  the  same  scale  the  amplitude  of  the  resulting  ray, 
and  the  angle  included  between  it,  and  either  side  will  represent  the  difference  of  phases  between  it  and 
the  ray  corresponding  ;  or,  which  comes  to  the  same  thing,  the  difference  of  their  origins  (when  reduced  to 
space.) 


LIGHT.  459 

Light.          Carol.  3.  Thus  in  the  case  of  complete  discordance,  the  diagonal  of  the  parallelogram  vanishes,  and  the  angle    Parl  "!• 
•— - v"~   becomes  180°,  or  half  a  circumference,  corresponding  to  a  difference  of  origins  of  half  an  undulation.     In  that  v-"' •v""'' 
of  complete  accordance,  the  angle  is  0,  or  360°,  and  the  origins  of  the  rays  coincide,  or  (which  comes  to  the       614. 
same  thing)  differ  by  an  exact  undulation,  and  the  diagonal  is  double  of  the  side,  so  that  the  intensity  of  the  c 
compound  ray  is  four  times  that  of  either  ray  singly.  concord  and 

Carol.  4.     If  the  origins  of  two  equally  intense   rays  differ  by  one  quarter  of  an  undulation,  the  resultant  discord. 
ray  will  have  its  amplitude  to  that  of  either  component  one,  as  v  2  :  1,  and,  therefore,  its  intensity  double,  and        615. 
its  origin  will  differ  one-eighth  of  an  undulation  from  that  of  either.     Thus  in  this  particular  care,  the  brightness 
of  the  compound  ray  is  the  sum  of  the  brightnesses  of  the  components,  and  its  position  exactly  intermediate  a  nuarter  of 
between  them.  an  undula- 

Corol.  5.     Any  ray  may  be  resolved  into  two,  differing  in  origin  and  amplitude,  by  the  same  rules  as  govern  lion, 
the  resolution  of  forces  in  Mechanics.  616. 

Coral.  6.     The  sum  of  the  intensities  of  the  component  rays  exceeds  that  of  the  resultant,  when  their  origins  Composi- 
differ  by  less  than  a  quarter  of  an  undulation,  falls  short  of  it  when  the   difference  is  between  J  and  £,  again  Button 
exceeds  it  when  between  \  and  ^,  and  so  on.     For  the  value  of  A',  above  found,  gives  rayS. 

o4  +  a'»  -  A2  =  2  a  a',  cos  k  ;  617. 

now  of,  a'4,  and  A2,  represent  the  intensities  of  the  respective  rays  whose  momenta  are  a,  a1,  and  A.  intensities 

Carol.  7      In  the  same  manner  may  any  number  of  similar  rays  be  compounded,  and  the  resultant  ray  will  be  of  simple 

similar  to  the  elementary  rays,  and  vice  versa.  anci  com- 

Let  us  now  consider  the  interference  of  waves  having  the  same  period  (or  colour)  but  in  all   other  respects  P0""1*  ray- 
dissimilar.  General' 

The  law  of  vibration  of  the  molecules  of  the  luminous  bodies  which  agitate  the  ether,  restricting  their  motions  problem  of 
to  ellipses  performed  in  planes,  the  same  will    hold  good  of  the  motions  of  each  molecule  of  the  ether.     Now  inter- 
every  elliptic  vibration,  or  rather  revolution,  performed  under  the  influence  of  a  force  directed  to  its  centre  and  ferences. 
proportional  to  the  distance,  is  decomposed  into  three  rectilinear  vibrations,  lying  in   any  three  planes  at  right 
angles  to  each  other,  each  of  which  separately  would  be  performed  by  the  action  of  the  same  force  in  the  same 
time,  and  according  to  the  same  laws  of  velocity,  time,  and   space.     Hence,  every  elliptic  vibration  may  be 
expressed  by  regarding  the  place  of  the  vibrating  molecule  at  any  instant  t  as  determined  by  three  coordinates 
r,  y,  z,  such  that,  0  being  an  arc  proportional  to  the  time,  we  shall  have 

dx 

x  ~  a  .  cos  (0  -f-  p)  ; ——  =  u  =  a. .  sin  (6  -j-  p) 

CL  t 

dy 

-  t»  =  0  .  sin  (0  +  g)    \  (2.) 


dt 

z  =  c .  cos  (0  -\-  r)  ;    —  — —  =  w  =  7  .  sin  (9  4-  r) 
a  t 

In  fact,  if  we  multiply  the  first  of  these  equations  by  an  indeterminate  I,  the  second  by  m,  and  the  third  by  n 
and  add,  we  get 

(3)  ;  I  x  -f-  my  -f-  rc  z  =  cos  0  {  I  a  .  cosp  -\-rnb  .  cos  q  -f-  n  c  .  cos  r  } 

—  sin  0  {  I  a  .  sin  p  -\-  m  b  .  sin  q  -j-  n  c .  sin  r  } 
and,  therefore,  if  we  determine  I,  m,  n,  so  that 

/  a  .  cos  p  -\-rnb  .  cos  q  -f-  n  c  .  cos  r  =  0  ;  /  a  .  sin  p  -f-  mb  .  sin  q  -j-  n  c  .  sin  r  =  0 
which  (being  equations  of  the  first  degree  only)  is  always  possible,  we  shall  have,  independently  of  0, 

and  this,  being  the  equation  of  a  plane,  shows  that  the  whole  curve  represented  by  the  above  equations  lies  in 
one  plane.     Again,  if  we  eliminate  0  between  the  equations,  involving  x  and  y  only,  we  have 

-  1     x  -  1    y 

3  n/i..  ** 


COS  --  cos  -y-  =  p  —  q, 


or,  taking  the  cosines  on  both  sides, 


and  reducing,  we  get  the  equation 

-fJ-  2.^-  4  .  cos  (p  -  9)  =  sin  (p  -  ?)»;        (5,) 

which  is  the  equation  of  an  ellipse  having  the  origin  of  the  x  and  y  in  its  centre,  and  the  same  is  true  mutatis 
mutandis  of  the  equations  between  x  and  z,  and  between  y  and  z.  Thus  the  curve  represented  by  the  three 
equations  between  x,  y,  z,  0,  has  an  ellipse  about  the  centre  for  its  projection  on  each  of  the  planes  at  right 
angles  to  each  other,  and  is,  of  course,  itself  an  ellipse. 

3o2 


460 


LIGHT. 


Suppose  now  two  systems  of  waves,  or  two  rays  coincident  in  direction,  to  interfere  with  each  other.     If  we    P«t  Iir. 
accent  the  letters  of  the  above  expressions  to  represent  corresponding  quantities  for  the  second  system,   we 
shall  have 


X  =  x  +  if  =  a  .  cos  (0  +  p)  -f  a'  .  cos  (9  -f  p') 
Y  =  y  +  y'  =  b  .  cos  (0  +  q)  +  V  .  cos  (0  +  g1) 
Z  =  z  -f  z'  —  c  .  cos  (0  +  r)  +  c'  .  cos  (0  -f  /) 


(6) 


and  similarly  for  the  velocities  u  -j-  u',  v  -f-  »',  w  +  «/.     In  the  same  manner,  then,  as  we  proceeded  in  the  case 
of  two  similar  rays,  let  us  suppose 

a  .  cos  (0  -f  p)  -j-  a'  .  cos  (0  -f  p1)  =  A  .  cos  (0  -J-  P) 
and  developing 

(a  .  cos  p  -{-  a'  •  cos  p1)  cos  0  —  (a  .  sin  p  -j-  a!  .  sin  pO  sin  0  —  A  .  cos  P  .  cos  0  —  A  .  sin  P  .  sin  6, 
whence  we  get  , 

p a  .  sinp  -f-  a!  .  sinp'  a  .  sinp  -f-  a!  .  s\npf     \ 

a  .  cosp  -j-  a' .  cosp'  '  sin  P  /  '     (7) 


or, 


A  =  */  a*  4-2  a  a1,  cos  (p  -  p')  +•«'• 


Thus  we  have  X  =  A  .  cos  (0  -f-  P),  and,  similarly,  Y  =  B  .  cos  (0  -f-  Q),  and  Z  =  C  .  cos  (0  +  R),  and  a  process 
exactly  similar  gives  us  the  corresponding  expressions  for  the  velocities. 

Thus  we  see  that  the  same  rules  of  composition  and  resolution  apply  to  dissimilar  as  to  similar  vibrations. 
Each  vibration  must  first  be  resolved  into  three  rectilinear  vibrations  in  three  fixed  planes  at  right  angles  to  each 
other.  These  must  be  separately  compounded  to  produce  new  rectilinear  vibrations  in  the  coordinate  planes, 
which  together  represent  the  resulting  elliptic  vibration,  and  will  have  the  same  period  as  the  component  ones. 
By  inverting  the  process,  a  vibration  of  this  kind  may  be  resolved  into  any  number  of  others  we  please,  having 
the  same  period. 

A  great  variety  of  particular  cases  present  themselves,  of  which  we  shall  examine  some  of  the  principal.  And 
first,  when  the  interfering  vibrations  are  both  rectilinear. 

Since  the  choice  of  our  coordinate  planes  is  arbitrary,  let  us  suppose  that  of  the  x,  y  to  be  that  in  which  both 
tne  vibrations  are  performed.  Of  course  the  resulting  one  will  be  performed  in  the  same.  Therefore  we  may 
put  z  =  0,  or  c  =  0,  </  =:  0,  and  content  ourselves  with  making 


;  y  =  6.cos(0-f-p) 
"  =  af.cos(&+p');  y'  =  b>  .  cos 


•) 
J  ' 


620. 

Composi- 
tion.  and 

tions  gene- 
rally. 

621. 
Case  of  in- 

of  rectilT 

nearvibra- 

tions. 


The  reiul- 

tant  vibra- 

tion  is,the 

general 

elliptic. 


622. 

Case  when 
the  resul- 
tant  is  rec- 

tiljnear- 


There  are,  therefore,  two  cases,  and  two  only  in  which  the  resulting  vibration  is  rectilinear.     The  first,  when 
p  _  p'  —  0,  or  when  the  component  vibrations  have  a  common  origin,  or  are  in  complete  accordance  ;  the 

Case  when  other,  when  —  =  -^-,  that  is,  when  they  are  both  performed  in  one  plane,  and  in  the  same  direction.     For  if 

tio^s  coin-    we  call  m  and  m1  the  amplitudes,  and  ^,  -f'  the  angles  they  make  with  the  axis  of  the  x,  we  have 
cide-  a  =  m  .  cos  ^  ;  b  —  m  .  sin  ^  ;  a1  =  m1  .  cos  ^  ;  b'  =  m'  .  sin  f-', 

so  that  the  above  equation  is  equivalent  to  tan  ^  =  tan  Y'1,  or  ^  =  Yr'- 
623.          The  latter  case  we  have  already  fully  considered.     In  the  former,  we  have  cos  (p  -  p')  =  0,  and,  therefore, 

A  =  a-|-  a';  B  =  &+&';  P=p;  Q=p, 


because  --  and  —j  are  constant  in  this  case,  and  X,  Y,  A,  B,  P,  Q,  denoting  as  in  the  general  case,  we  have 

X  =  A.  cos(0  +  P);    Y=B.cos(0  +  Q); 
and,  by  elimination  of  0, 

=  sin(P      Q)1;  (9) 

where  A,  B,  P,  Q,  are  determined  as  in  equations,  (7.)     In  the  general  case,  then,  the  resulting  vibration  is 
elliptic. 

The  ellipse  degenerates  into  a  straight  line  by  the  diminution  of  its  minor  axis  when  P  =  Q.     Now  this  gives 
tan  P  =  tan  Q,  or 

a.  sin  p-j-o'.  sinp'      _   6  .  sinp  -j-b' .  sinp7 

a.  cosp-)-  a' .  cosp1          6.  cosp  -{-b' .  cosp' 
which,  reduced,  takes  the  form 


LIGHT.  401 

IJ*ht-  Y          6+6'  pall  in. 

~^~-S  and,  finally,  —  = —  =  tan  0     ;  (10)  , 

X          a  -j-  a1  TT^s^ 

which  is  the  tangent  of  the  angle  made  by  the  resulting1  rectilinear  vibration  with  the  axis  of  the  x.  complete 

If  we  put  M  for  the  amplitude  of  the  resulting  vibration,  we  have  M  .  cos  0  =  A ;  M  .  sin  0  =  B  ;  therefore,  accordance 
M» .  (cos  0s  +  sin  0s)  or  M«  =  A*  +  B".  of  non- 

coincident 

Now,  A8  =  (a  -j-  a')8  =  (m .  cos  ^  -f-  m  .  cos  ^- )8  vibrations. 

B2  =  (6  4-  6')'  =  (m  .  sin  \!r  +  m'  .  sin  -f ')'  624- 

Amplitude 
and,  therefore,  adding  these  values  together,  and  reducing  and  position 

M'  =  m8  +  2  m  m' .  cos  (^  -  +')  +  m'8 ;         (11)  °L™L. 

Now,  ijr  —  ijr'  is  the  angle  between  the  directions  of  the  component  vibrations,  so  that  this  equation  expresses  tion  deter- 
that  the  amplitude  of  the  resultant  vibration  is  in  this  case  also  the  diagonal  of  a  parallelogram,  whose  sides  mined- 

are  the  amplitudes  of  the  component  ones ;  and  it  is  easily  shown,  by  substituting  in  tan  0  =  —  the  above 

a-\-  a! 
values  of  a  -f-  a',  6  -f-  6',  that  the  diagonal  has  also  the  position  of  the  resultant  line  of  vibration. 

Carol.  1.     Any  rectilinear  vibration  may  be  resolved  into  two  other  rectilinear  vibrations,  whose  amplitudes      625. 
are  the  sides  of  any  parallelogram,  of  which  the  amplitude  of  the  original  vibration  is  the  diagonal,  and  which 
are  in  complete  accordance,  or  have  a  common  origin  with  it. 

Carol.  2.     Hence  any  rectilinear  vibration   may  be  readily  reduced  to  the  directions  of  two   rectangular      626. 
coordinates,  or,  if  necessary,  into  those  of  three,  by  the   rules  of  the  resolution  of  forces,  and  the  component 
vibrations,  however  numerous,  will  be  in  complete  accordance  with  the  resultant. 

The  ellipse  degenerates  into  a  circle  when  cos  (P  —  Q)  =  0,  or  P  —  Q  =  90°,  and,  also,  A  =  B.     Now  the      $27 

former  condition  gives  tan  P  -f-  cot  Q  =  0,  that  is  Case  of ' 

a  .  sin  p  -4-  a  .  sin  p'          b  .  cos  p  -4-  6' .  cos  r/  circular 

• ^-1 > ^T  +  -7 .  ./ ^-T  —  °  vibrations. 

a  .  cosj»  -f-  a  .  cosp  6  .  sin  p  -j-  6'  .  smp' 


a  b  +  a'  V  .    nf  .  sin  2  y*  -f  m'2  sin  2 

cos  (p  -  p)  =  -         ,         ,      =  -  4  - 


or  reducing 


The  condition  A  =  B,  or  A8  =  B8,  gives 

a1  +  2  a  a  .  cos  (p  -  p')  +  a'8  =  b*  -f  2  b  b'  .  cos  (p  -  pf)  -f  6'8 
whence  we,  in  like  manner,  obtain 

,    _          (a8  -f-  a'1)  —  (68  -f-  6'«)   _          i    m*  •  cos  2  ty  +  OT"  •  CO9  2  Y"' 

and,  equating  the  values  of  cos  (p  —  p'),  we  find  the  following  relation  between  a,  a',  6,  V,  which  must  subsist 
when  the  vibrations  are  circular, 

/    „  „!  \ 

f  ft«  _  a'1  -  b'>)  =  0. 

The  vanishing  of  the  first  factor  gives  no  circular  vibration,  it  being  introduced  with  the  negative  root  of  the 
equation  A8  =  B8,  with  which  we  have  no  concern.  The  other  gives 

a8  -f-  68  =  a'8  -f-  6'8,  or  m  —  m', 

which  shows  that  the  component  vibrations  must  have  equal  amplitudes.  Now,  if  for  a  and  6  we  write  their 
values  m  .  cos  fy  and  m  .  sin  T^-,  and  for  a'  and  b',  respectively,  m  .  cos  ijr'  and  m  .  sin  ty',  in  either  of  the 
expressions  for  cos  (p  —  p'),  it  will  reduce  itself  to 

cos  (p  —  jT)  =  —  cos  (^  —  ty')  ;  or,  p  —  p'  =  180°  —  (^  —  Y'O- 

Hence  it  appears,  that  the  interference  of  two  equal  rectilinear  vibrations  will  produce  a  resultant  circular  one, 
provided  the  difference  of  their  phases  be  equal  to  the  supplement  of  lue  angle  their  directions  make  with  each 
other,  so  that  when  the  molecule  is  just  commencing  its  motion  towards  its  centre,  in  virtue  of  one  vibration,  it 
shall  be  receding  from  it  at  an  obtuse  angle  with  this  motion,  in  virtue  of  the  other. 

Carol.      Hence,  if  two  vibrations  have  equal  amplitudes,  but  differ  in  their  phases  by  a  quarter  of  an  undula- 
tion, their  resultant  vibration  will  be  circular. 

We  are  now  in  a  condition  to  explain  what  becomes  of  the  portions  of  the  secondary  waves  which  diverge      62S. 
obliquely  from  the  molecules  of  the  primary  ones,  as  alluded  to  in  Art.  595,  and  to  explain  the  mode  in  which  Fig.  130 
those  which  do  not  conspire  with  the  primary  wave  mutually  destroy  each  other.     To  this  end,  conceive  the  sur- 
face of  any  wave  A  B  C  to  consist  of  vibratory  molecules,  all  in  the  same  phase  of  their  vibrationn.     Then  will  the 
motion  of  any  point  X  (fig.  130)  be  the  same,  whether  it  be  regarded  as  arising  from  the  original  motion  of  S,  Mutml 
or  as  the  resultant  of  all  the  motions  propagated  to  it  from  all  the  points  of  this  surface.     Conceive  the  surface  destruction 
ABC  divided  into  an  infinite  number  of  elementary  portions,  such  that  the  difference  of  distance  of  each  con-  °f  secon~ 
secutive  pair  from  X  shall  be  constant,  or  =  d  f,  putting  the  distance  of  any  one  from  that  point  =  /,-  and  let 
A  B,  B  C,  CD,  &c.,  and  A  6,  6  c,  c  d,  &c.  be  finite  portions  of  the  surface  containing  each  the  same  number  of 


462  L  I  G  H  T. 

Light.  these  elements,  and  in  each  of  which  the  corresponding  values  of  f  are  exactly  half  an  undulation  (J  X)  greater  Part 
v—  'v""-''  than  in  the  preceding,  so  that  (for  instance)  B  X  =  A  X  +  \  X,  C  X  =  B  X  -f  \  X,  &c.  Then  it  is  evident,  '•—  -  v 
that  the  vibrations  which  reach  X  simultaneously  from  the  corresponding  portions  of  any  two  consecutive  ones, 
as  of  A  B  and  B  C,  will  be  in  exactly  opposite  phases  ;  and,  therefore,  were  they  of  equal  intensity,  and  in 
precisely  the  same  direction,  would  interfere  with,  and  destroy  each  other.  Now,  first,  with  regard  to  their 
intensity,  this  depends  on  the  magnitudes  of  the  elements  of  the  wave  A  B,  from  which  they  are  derived,  and  on 
the  law  of  lateral  propagation.  Of  the  latter,  we  know  little,  a  priori  ;  but  all  the  phenomena  of  light  indicate 
a  very  rapid  diminution  of  intensity,  as  the  direction  in  which  the  secondary  undulations  aie  propagated  deviates 
from  that  of  the  primary.  With  respect  to  the  former,  it  is  evident  that  the  elements  in  the  immediate  vicinity 
of  the  perpendicular  A  X,  corresponding  to  a  given  increment  d  f  of  the  distance  from  X,  are  much  larger  than 
those  remote  from  it  ;  so  that  all  the  elements  of  the  portion  A  B  are  much  larger  than  those  in  B  C,  and  these 
again  than  in  those  of  C  D,  and  so  on.  Thus  the  motion  transmitted  to  X  from  any  element  in  A  B  will  be 
much  greater  than  that  from  the  corresponding  one  in  B  C,  and  that  again  greater  than  that  from  the  element  in 
C  D,  and  so  on.  Thus  the  motion  arriving  at  G,  from  the  whole  series  of  corresponding  elements,  will  be  repre- 
sented by  a  series  such  as  A  —  B  +  C  —  D  +  E  —  F+  &c.,  in  which  each  term  is  successively  greater  than 
that  which  follows.  Now  it  is  evident  that  the  terms  approach  with  great  rapidity  to  equality  ;  for  if  we  consider 
any  two  corresponding  elements  as  M,  N  at  a  distance  from  A  at  all  considerable,  the  angles  X  M  and  X  N  make 
with  the  surface  approach  exceedingly  near  to  equality,  so  that  the  obliquity  of  the  secondary  wave  to  the  pri- 
mary, and  of  course  its  intensity,  compared  with  that  of  the  direct  wave,  is  very  nearly  alike  in  both  ;  and  the 
elements  M,  N  themselves,  at  a  distance  from  the  perpendicular,  approach  rapidly  to  equality,  for  the  elementary 
triangles  M  mo,  M  np  are  in  this  case  very  nearly  similar,  and  have  their  sides  m  o,  np  equal  by  hypothesis. 
Finally,  the  lines  M  X,  N  X  approach  nearer  to  each  other  in  direction  so  as  to  produce  a  more  complete  inter- 
ference, as  their  distance  from  A  is  greater. 

629.  Thus  we  see  that  the  terms  of  the  series  A  —  B  +  C  —  D  -(-  &c.,  at    a   distance  from   its    commencement, 
have   on     all    accounts    (viz.    their   smallness,    near    approach    to    equality,  and  disposition  to  interfere)    an 
extremely  small  influence  on  its  value  ;  and  as  the  same  is  true  of  every  set  of  corresponding  elements  into 
which  the  portions  A  B,  B  C,  &c.  are  divided,  it  is  so  of  their  joint  effect,  so  that  the  motion  of  the  molecule  X  is 
governed  entirely  bv  that  of  the  portion  of  the  wave  ABC  immediately  contiguous  to  A,  the  secondary  vibrations 
propagated  from  parts  at  a  distance  mutually  interfering  and  destroying  each  others  effect. 

630.  It    is  obvious,    that   in    the    case    of    refraction    or    reflexion,  we    may  substitute  for   the  wave  AM  the 
refracting  or  reflecting  surface  ;  and  for  the  perpendicular  X  A  the  primary  refracted  ray,  when  the  same  things, 
mutatis  mutandis,  will   hold    good.     See  M.  Fresnel's  Paper   entitled  Explication  de  la  Refraction  dans   le 
Systeme  des  Ondes,  published  in  the  Bulletin  de  la  Societe  Philomatique,  October,  1821. 

gg^  This  is  the  case  when  the  portion  of  the  wave  A  B  C  D  whose  vibrations  are  propagated  to  X  is   unlimited, 

Case  of  a     or  a'  'east  so  considerable,  that  the  last  term  in  the  series  A  —  B  +  C  —  &c.  is  very  minute  compared  with  the 

wave  first.     But  if  this  be  not  the  case,  as,  if  the  whole  of  a  wave  except  a  small  part  about  A  be  intercepted  by  an 

transmitted  obstacle,  the  case  will  be  very  different.    It  is  easy  on  this  supposition  to  express  by  an  integral  the  intensity  of  the 

tiinmgh  a     undulatory  motion  of  X,  compared  with  what  it  would  be  on  the  supposition  of  no  obstacle  existing.     For  this 

[  purpose,  let  d2  *  be  the  ifmiitude  of  any  vibrating  element  of  the  surface,  f  its  distance  from  X  =  M  X,  and  let 

0(0)  be  the  function  of  the  a.V  made  by  a  laterally-divergent  vibration  with  the  direct  one,  which  expresses  its 

relative  intensity,  and  which  is  unity  when  0  =  0,  and  diminishes  with  great  rapfdity  as  0  increases.     Then  if  t 

be  the  time  since  a  given  epoch,  X  =   the  length  of  an  undulation,  S  A  =  a,  the  phase  of  a  vibration  arriving 

at  X  by  the  route  S  M  X  will  be  2  TT  (  —  —  J  ,  and  the  velocity  produced  in  X  thereby  will  be  repre- 

sented by  a  .  d*  s  .  0  (0)  .  sin  2  TT  (  -—  --  -  ±-si  J,  so  that  the  whole  motion  produced  will  be  represented  by 

//  a  .  d*s  .  0  (0)  .  sin  2  ir  |-1  -   -iyZ  J 
the  integral  being  extended  to  the  limits  of  the  aperture. 

632  Carol  1     If  but  a  very  small  portion  of  the  wave  be  permitted  to  pass,  as  in  the  case  of  a  ray  transmitted 

through  a  very  small  hole,  and  received  on  a  distant  screen,  0  and  0  (0)  are  very  nearly  constant,  so  that  i 
motion  excited  in  X  is  in  this  case  represented  by 


We  shall  have  occasion  to  revert  to  these  expressions  hereafter. 

§  IV.  Of  the  Colours  of  Thin  Plates. 

(533  Every  one   is  familiar  with  the  brilliant  colours  which  appear  on  soap-bubbles  ;   with  the   iridescent    Hues 

General  produced  by  heat  on  polished  steel  or  copper  ;  with  those  fringes  of  beautiful  and  splendid  colours  which  appear 
account  of  jn  the  cracks  of  broken  glass,  or  between  the  laminae  of  fissile  minerals,  as  Iceland  spar,  mica,  sulphate  of 
the  pheno-  lime>  &c  jn  an  t|lese>  anc[  an  innnile  variety  of  cases  of  the  same  kind,  if  the  fringes  of  colour  be  examined 


LIGHT.  463 

Light.      with  care  they  will  be  found  to  consist  of  a  regular  succession  of  hues,  disposed  in  the  same  order,  and  deter-    Part  III. 
— v"^''  mined,  obviously,  not  by  any  colour  in  the  medium  itself  in  which  they  are  formed,  or  on  whose  surfaces  they  '_ -  v  •_- 
appear,  but  solely  by  its  greater  or  less  thickness.     Thus  a  soap-bubble  (defended  from  currents  of  air  by  beino- 
placed  under  a  glass)  at  first  appears  uniformly  white  when  exposed  to  the  dispersed  light  of  the  sky  at  an  open 
window ;  but,  as  it  grows  thinner   and  thinner  by  the  subsidence  of  its  particles,  colours  ben~in  to  appear  at  its 
top  where  thinnest,  which  grow  more  and  more  vivid,  and  (if  kept  perfectly  still)  arrange  themselves  in  beautiful 
horizontal  zones  about  the  highest  point  as  a  centre.     This  point,  when  reduced  to  extreme  tenuity,  becomes 
black,  or  loses  its  power  of  reflecting  light  almost  entirely.     After  which  the  bubble  speedily  bursts,  its  cohesion 
at  the  vertex  being  no  longer  sufficient  to  counteract  the  lateral  attraction  of  its  parts. 

But  as  it  is  a  matter  of  great  delicacy  to  make  regular  observations  on  a  thing  so  fluctuating  and  unmanao-e-      634. 
able  as  a  soap-bubble,  the  following  method  of  observing  and  studying  the  phenomena  is  far  preferable.     Let  a  Ri"?s 
convex  lens,  of  a  very  long  focus  and  a  good  polish,  be  laid  down  on  a  plane  glass,  or  on  a  concave  glass  lens  (nrmelli  be- 
having a  curvature  somewhat  less  than  the  convex  surface  resting  on  it ;  so  that  the  two  shall  touch  in  but  a  lv 
single  point,  and   so  that  the  interval  separating  the  surfaces  in  the  surrounding  parts  shall  be  exceedingly  g°iscs. 
small.     If  the  surfaces  be  very  carefully  cleaned  from  dust  before  placing  them  together,  and  the  combination  be 
laid  down  before  an  open  window  in  full  daylight,  the  point  of  contact  will  be  seen  as  a  black  spot  in  the  general 
reflexion  of  the  sky  on  the  surfaces,  surrounded  with  rings  of  vivid  colours.     A  glass   of  10    or  12  feet  focus 
laid  on  a  plane  glass,  will  show  them  very  well.     If  one  of  shorter  focus  be  used,  the  eye  may  be  assisted  by  a 
magnifying  glass.     The  following  phenomena  are  now  to  be  attended  to  : 

Phenomenon  1.    The  colours,  whatever  glasses  be  used,  provided  the  incident  light  be  white,  always  succeed      635. 
each  other  in  the  very  same  order ;  that  is,  beginning  with  the  central  black  spot,  as  follows  :  Order  of 

First  ring,  or  first  order  of  colours, — black,  very  faint  blue,  brilliant  white,  yellow,  orange,  red.  succession 

Second  ring,  or  second  order, — dark  purple  or  rather  violet,  blue,  green,  (very  imperfect,  a  yellow-green,)  of  the 
vivid  yellow,  crimson  red. 

Third  ring,  or  third  order, — purple,  blue,  rich  grass  green,  fine  yellow,  pink,  crimson. 
Fourth  ring,  or  fourth  order, — green,  (dull  and  bluish,)  pale  yellowish  pink,  red. 
Fifth  ring,  or  fifth  order, — pale  bluish  green,  white,  pink. 
Sixth  ring,  or  sixth  order, — pale  blue-green,  pale  pink. 

Seventh  ring,  or  seventh  order, — very  pale  bluish  green,  very  pale  pink.     After  these,  the  colours  become  so 
faint  that  they  can  scarcely  be  distinguished  from  white. 

On  these  we  may  remark,  that  the  green  of  the  third  order  is  the  only  one  which  is  a  pure  and  full  colour,  that  of  636. 
the  second  lieing  hardly  perceptible,  and  of  the  fourth  comparatively  dull  and  verging  to  apple  green  ;  the  yellow 
of  the  second  and  third  order  are  both  good  colours,  but  that  of  the  second  is  especially  rich  ancfsplen'did  ;  that  of 
the  first  being  a  fiery  tint  passing  into  orange.  The  blue  of  the  first  order  is  so  faint  as  to  be  scarce  sensible, 
that  of  the  second  is  rich  and  full,  but  that  of  the  third  much  inferior;  the  red  of  the  first  order  hardly  deserves 
the  name,  it  is  a  dull  brick  colour;  that  of  the  second  is  rich  and  full,  as  is  also  that  of  the  third;  but  they  all 
verge  to  crimson,  nor  does  any  pure  scarlet,  or  prismatic  red,  occur  in  the  whole  series. 

Phenomenon  2.  The  breadths  of  the  rings  are  unequal.    They  decrease,  and  the  colours  become  more  crowded,      637. 
as  they  recede  from  the  centre.     Newton  (to  whom  we  owe  the  accurate  description  and  investigation  of  their  J^Vf  th<j 
phenomena)  found  by  measurement  the  diameters  of  the  darkest  (or  purple)  rings,  just  when  the  central  black  the  rinw  ° 
spot  began  to  appear  by  pressure,  and  reckoning  it  as  one  of  them  to  be  as  the  square  roots  of  the  even  numbers  and  tlnck- 
0,  2,  4,  6,  &c. ;  and  those  of  the  brightest  parts,  of  the  several  orders  of  colours,  to  be  as  the  square  roots  of  the  nesses  at 
odd  numbers   1,  3,   5,   7,  &c.     Now  the  surfaces  in  contact  being  spherical,  and  their  radii  of  curvature  very  which  they 
great  in  proportion  to  the  diameters  of  the  rings,  it  follows  from  this  that  the  intervals  between  the  surfaces  at  appea 
the  alternate  points  of  greatest  obscurity  and  illumination  are  as  the  natural  numbers  themselves  0,  1,2,  3,  4, 
&c.     The  same  measurements,  when  the  radii  of  curvature  of  the  contact  surfaces  are  known,  give  the  absolute 
magnitudes  of  the  intervals  in  question.     In  fact,  if  r  and  r1  be  the  curvatures  of  two  spherical  surfaces,  a  convex 
and  concave,  in  contact,  and  D  the  diameter  of  any  annulus  surrounding  their  point  of  contact,  the  interval  of 
the  surfaces  there  will  be  the  difference  of  the  versed  sines  of  the  two  circular  arcs  having  a  common  chord  D. 
Now  (fig.  130)  if  A  E  be  the  diameter  of  the  convex  spherical  surface  A  D,  we  have  EA  :  A  D  ; ;  A  D  :  D  B 

AD2         D2  D-  1 

.  . ,     =  — —  r,  and  in  like  manner  B  C  =   -   —  r',   so  that——  D*  (r  —  r')  =  DC,  the  interval  of  the 
A  &  y  h  8 

surfaces  at  the  point  D.  Thus  Newton  found,  for  the  interval  of  the  surfaces  at  the  brightest  part  of  the  first 
ring,  one  178000dth  part  of  an  inch  ;  and  this  distance,  multiplied  by  the  even  natural  numbers  0,  2,  4,  6,  8,  &c. 
gives  their  distance  at  the  black  centre  and  the  darkest  parts  of  the  purple  rings,  and  by  the  odd  ones  1,  3,  5,  &c. 
their  intervals  at  the  brightest  parts. 

Phenomenon  3.  If  the  rings  be  formed  between  spherical  glasses  of  various  curvatures,  they  will  be  found  to      638. 
be  larger  as  the  curvatures  are  smaller,  and  vice  vend ;  and  if  their  diameters  be  measured  and  compared  with  Invariable 
the  radii  of  the  glasses,  it  will  be  found,  that,  provided  the  eye  be  similarly  placed,  the  same  colour  is  invariably  re'atio"  '"' 
produced  at  that  point,  or  that  distance  from,  the  centre  where  the  interval  between  the  surfaces  is  the  same.  *™ j**" 
Thus,  the  white  of  the  first  order  is  invariably  produced  at  a  thickness  of  one  178000th  of  an  inch  ;  the  purple,  thickni 
which  forms  the  limit  of  the  first  and  second  orders,  at  twice  that  thickness.     So  that  there  is  a  constant  rela-  of  plates. 
tion  between  the  tint  seen  and  the  interval   of  the  surface  *  where  it  appears.     Moreover,  if  the  glasses   be 
distorted  by  violent   and   unequal  pressure,    (as  is  easily  done  if  thin  lenses  be  used,)  the  rings   lose   their 
circular  figure,  and  extend  themselves  towards  the  part  where  the  irregular  pressure  is  applied,  so  as  to  form  a 
species  of  level  lines   each  marking  out  a  series  of  points  where  the  surfaces  are  equidistant.     Thus,  too,  if  a 


s  an«. 
es^es 


464  LIGHT. 

Light,      cylinder  be  laid  on  a  plane,  the  rings  pass  into  straight  lines  arranged  parallel  to  its  line  of  contact,  but  following    Part  II. 
—  »"v—-  -  the  same  law  of  distance  from  that  line  as  the  rings  from  their  dark  centre,  and  if  the  glasses  be  of  irregular  -_^v-^. 
curvature,  as  bits  of  window  glass,  the  bands  of  colour  will   follow  all  their  inequalities  ;    yet    more,  if  the 
pressure  be  very  cautiously  relieved,  so  as  to  lift   one  glass  from  the   other,  the  central  spot  will   shrink  and 
disappear,  and  so  on  ;    each  ring  in  succession  contracting  to  a  point,  and  then  vanishing,  so  as  to  bring  all  the 
more  distant  colours   successively  to  the  centre,  as  the  glasses  recede  from  absolute  contact.     From  all  these 
phenomena  it  is  evident,  that  it  is  the  distance  between  the  surfaces  only  at  any  point  which  determines  the 
colour  seen  there. 

639.  Phenomenon  4.     This  supposes,  however,  that  we  observe  them  with  the  eye  similarly  placed,  or  at  the  same 

KH't-ct  of  angle  of  obliquity.  For  if  the  obliquity  be  changed  by  elevating  or  depressing  the  eye,  or  the  glasses,  the 
obliquity  of  diameters  (but  not  the  colours)  of  the  rings  will  change.  As  the  eye  is  depressed,  the  rings  enlarge  ;  and  the 
ice-  same  tint  which  before  corresponded  to  an  interval  of  the  178000th  of  an  inch,  now  corresponds  to  a  greater 


interval.  This  distance  (-1-,^?™)  is  determined  by  measures  taken  nearly  at,  and  reduced  by  calculation  exactly 
to,  a  perpendicular  incidence.  At  extreme  obliquities,  however,  the  diameters  of  the  several  rings  suffer  only  a 
certain  finite  dilatation,  and  Newton's  measures  led  him  to  the  following  rule  :  viz.  "  That  the  interval  between 
the  surfaces  at  which  any  proposed  tint  is  produced,  is  proportional  to  the  secant  of  an  angle  whose  sine  is  the 
first  of  106  arithmetical  mean  proportionals  between  the  sines  of  incidence  and  refraction,  into  the  glass  from  the 
air,  or  other  medium  included  between  the  surfaces,  beginning  with  the  greater  ;"  or,  in  algebraic  language,  the 
relative  index  of  refraction  being  /*,  and  0  the  angle  of  incidence,  and  p  that  of  refraction  of  the  ray  as  it  passes 
out  of  the  rarer  medium  into  the  denser  ;  then,  if  t  be  the  interval  corresponding  to  a  given  tint  at  the  oblique 
incidence  0,  and  T  at  a  perpendicular  incidence,  we  shall  have 

/  =  T  .  sec  u  where  sin  u  =  sin  6  --  (sin  0  —  sin  p) 
but  sin  p  =  —  .  sin  0,  consequently  we  have 

106  -J-  — 
t  =  T  .  sec  u  ;  sin  u  =  -  —      ^     .  sin  0  .  =  -      ^  .  sin0. 

640.  To  see  the  rings  conveniently  at  extreme  obliquities,  a  prism  maybe  used,  laid  on  a  convex  lens,  as  in  fig.  132. 
Fig.  13:2.      If  the  eye  be  placed  at  K,  the  set  of  rings  formed  about  the  point  of  contact  E  will  be   seen  in   the  direction 
Kings  seen  K  H,  and  as  the  eye  is  depressed  towards  the  situation  I,  where  the  ray  IG  intromitted  from  I  would  just  begin 

rough  a  jo  suflfer  total  reflexion,  the  rings  are  seen  to  dilate  to  a  certain  considerable  extent.  When  the  eye  reaches  I, 
the  upper  half  of  the  rings  disappears,  being  apparently  cut  off  by  the  prismatic  iris  of  Art.  555,  which  is  seen 
in  that  situation,  but  the  black  central  spot  and  the  lower  half  of  the  rings  remains  ;  but  when  the  eye  is  still 
further  depressed  the  rings  disappear,  and  leave  the  central  spot,  like  an  aperture  seen  in  the  silvery  whiteness 
of  the  total  reflexion  on  the  base  of  the  prism,  and  dilated  very  sensibly  beyond  the  size  of  the  same  spot  seen 
in  the  position  K  H  :  thus  proving,  that  the  want  of  reflexion  on  that  part  of  the  base  extends  beyond  the  limits 
of  absolute  contact  of  the  glasses,  and  that,  therefore,  the  lower  surface  interferes  with  the  action  of  the  upper, 
and  prevents  its  reflexion  while  yet  a  finite  interval  (though  an  excessively  minute  one)  intervenes  between 
them.  Euler  has  made  this  an  objection  to  the  undulatory  theory,  but  the  objection  rests  on  no  solid  grounds, 
as  it  is  very  reasonable  to  conclude,  that  the  change  of  density  or  elasticity  in  the  ether  within  and  without  a 
medium  is  not  absolutely  per  saltum,  but  gradual.  If  so,  and  if  the  change  take  place  without  the  media,  the 
approach  of  two  media  within  that  limit,  within  which  the  condensation  of  the  ether  takes  place,  will  alter  the 
law  of  refraction  from  either  into  the  interval  separating  them. 

641.  In  order,  however,  to  see  to  the  greatest  advantage  the  colours  refl°"trd  by  a  plate  of  air  at  great  obliquities, 
Fringes        the  following  method,  first  pointed  out  by  Sir  William  Herschel,  may  be  employed.     On  a  perfectly  plane  glass 
seen  when  a  or  metallic  mirror,  before  an  open  window,  lay  an  equilateral   prism,  having  its  base  next   the  glass  or  mirror 
prism  is       very  truiy  piane>  and  looking  in  at  the  side  AC,  fig.  133,  the  reflected  prismatic  iris,  a,  b,  c,  will  be  seen  as  usual 
planeelass    'n  tne  d'rect'on  E  F,  where  a  ray  from  E  would  just  be  totally  reflected.     Within  this  iris,  and  arranged  parallel 
Fig.  133.      to  it,  are  seen  a  number  of  beautiful   coloured  fringes,  whose  number  and  distances  from  each  other  vary  with 

every  change  of  the  pressure  ;  their  breadths  dilating  as  the  pressure  is  increased,  and  vice  versa.  They  do  not 
require  for  their  formation,  that  the  surfaces  should  be  exceedingly  near,  being  seen  very  well  when  the  prism  is 
separated  from  the  lower  surfaces  by  the  thickness  of  thin  tissue  paper,  or  a  fine  fibre  of  cotton  wool  interposed, 
but  in  this  case  they  are  exceedingly  close  and  numerous.  If  the  pressure  be  moderate,  they  are  nejrly  equi- 
distant, and  are  lost,  as  it  were,  in  the  blue  iris,  without  growing  sensibly  broader  as  they  approach  it.  As  the 
intervals  of  the  surfaces  is  diminished,  they  dilate  and  descend  towards  the  eye,  appearing,  as  it  were,  to  come 
down  out  of  the  iris.  They  do  not  require  for  their  formation  a  perfect  polish  in  the  lower  surface.  An  emeried 
glass,  so  rough  as  'o  reflect  no  regular  image  at  any  moderate  incidence,  shows  them  very  well.  The  experi- 
ment is  a  very  easy  one,  and  the  phenomena  so  extremely  obvious  and  beautiful,  that  it  is  surprising  it  should 
not  have  been  noticed  and  described  by  Newton,  especially  as  it  affords  an  excellent  illustration  of  his  law 
above  stated  To  understand  this,  let  EH,  E  K,  E  L  be  any  rays  from  E  incident  at  angles  somewhat  less 
than  that  of  total  reflexion  on  the  base  ;  they  will  therefore  be  refracted,  and,  emerging  at  the  base  B  C,  will  be 
reflected  at  M  N,  (the  obliquity  of  the  reflexion  being  so  great,  that  even  rough  surfaces  reflect  copiously  and 
regularly  enough  for  the  purpose,  Art.  558,)  and  will  pursue  the  courses  H  D  Vp,  K  F  Q  7,  L  G  R  r,  &c.  entering 
the  prism  again  at  P,  Q,  R.  Reciprocally,  then,  rays  p  P,  q  Q,  &c.  incident  at  P,  Q,  &c.  in  these  directions, 


LIGHT.  465 

Light.       wiH  enter  the  eye  at  E  after  traversing  the  interval  R  C  N  M,  and  being  reflected  at  M  N,  and  will  affect  the  eye     1'art  III. 
—  -v'-"''  with  the  colour  corresponding  to  that  obliquity  and  that  interval  between  the  surfaces  which  is  proper  to  each.  >—  •  -v— 
If  then  we  put,  as  above,  0  for  the  exterior  angle  of  incidence  of  the  ray  D  H  on  the  base  of  the  prism,  and 
lake 

1  106/.+  1 

- 


the  tint  seen  in  the  direction  E  H  will  (abstraction  made  of  the  dispersion  at  the  surface  A  C)  be  the  same  with 
that  reflected  at  a  perpendicular  incidence,  by  a  plate  of  air  of  the  thickness  T  =  t  .  cos  u  =  t  */  1  —  If  .  slap1, 
where  t  =  the  distance  between  the  surfaces  B  C,  M  N.  There  will,  therefore,  appear  a  succession  of  colours 
in  the  several  consecutive  situations  of  the  line  E  II,  analogous  to  those  of  the  coloured  rings,  (except  in  so  far 
as  the  dispersion  of  the  side  A  C  alters  the  tints  by  separating  their  component  rays.) 

But  the  whole  series  of  colours  will  not  be  seen,  because  those  which  require  greater  obliquities  than  that  at      642. 
which  total  reflexion  takes  place,  cannot  be  formed.     In  fact,  the   angle,  reckoned   from  the  vertical  at  which  a 
tint  corresponding  to  a  thickness  T  in  the  rings  would  be  formed,  is  given  by  the  equation 


sin 


.    1      ./,       /TV_  214    ,/    '     /TV 

-T-V   -Viy^wV      (r)  ' 


o 

taking  p.  =  —  for  glass,  which  it  is  very  nearly.     Now,  according  to  this,  the  central  tint,  or  black  of  the  first 

order,  which  is  formed  when  T  =  C,  requires  that 

1  1 

Sln  p  =  —  = 


k  p.  -  1 

107 

which  being  greater  than  —  shows  that  this  tint  lies  above  the  situation  of  the  iris,  and  cannot  therefore  be 
seen.    The  first  visible  tint  will  be  that  close  to  the  iris,  where  sin  p  =  —  which  gives 


nearly,  or  .     Hence  it  appears,  that  these  fringes  would  be  seen,  by  an  eye  immersed  in  the  prism,  when 

the  interval  between    its  base  and  the  glass  it  rests  on  is  more  than  12  times  that  at  which  colours  are  formed 

13                          1 
at  a  perpendicular  incidence,  t.  e.  at  12'25  x ,  or  about- th  of  an  inch,  which  is  about  the  thickness 

of  fine  tissue  paper.  Moreover,  from  this  value  of  T,  we  see  that  the  first  tint  immediately  visible  below  the 
iris  ascends  in  the  scale  of  the  rings  (i.  e.  belongs  to  a  point  nearer  their  centre)  as  the  value  of  t  diminishes, 
or  as  the  prism  is  pressed  closer  to  the  glass  ;  and  this  explains  why  the  fringes  become  more  numerous,  and 
appear  to  come  out  of  the  iris  by  pressure.  With  regard  to  their  angular  breadth,  (still  to  an  eye  immersed  in  the 

J  incl 

prism.)     If  we  put  e  =  ,  we  have,  putting  pa,  pt,  &e.  for  the  values  of  p,  corresponding   to  the  several 

orders  of  visible  tints, 

1  1 

sin  p    =   —  ;    sin  nl  =  — 
P-  k 

very    nearly,    sin  p,  =   —    (1—0-079. )  and  so  on.     The  sines  then  of  the  incidences  at  which  the  several 

P     \  <•    J 

orders  of  colours  are  developed,  beginning  at  the  iris,  increase  in  arithmetical  progression,  so  that  the  fringes  must 
be  disposed  in  circular  arcs  parallel  to  the  iris,  and  their  breadths  must  be  nearly  equal,  and  greater  the  greater 
the  pressure  or  the  less  t  is,  all  which  is  conformable  to  observation.  The  refraction  of  the  side  of  the  prism 
between  the  eye  and  the  base,  however,  disturbs  altogether  the  succession  of  colours  in  the  fringes,  and 
in  particular  multiplies  the  number  of  visible  alternations  to  a  great  extent,  in  a  manner  which  will  be  evi- 
dent on  consideration.  We  have  been  rather  more  particular  in  explaining  the  origin  of  these  fringes,  and 
referring  them  to  the  general  phenomena  observed  by  Newton,  because  up  to  the  present  time  we  believe  no 
strict  analysis  of  them  has  been  given,  as  well  as  on  account  of  the  great  beauty  of  the  phenomenon  itself.  If 
we  hold  the  combination  up  to  the  light,  and  look  through  the  base  of  the  prism  and  the  glass  plate,  so  as  to 
see  the  transmitted  iris  of  Art.  556,  its  concavity  will,  in  like  manner,  be  seen  fringed  with  bands  of  colours  of 
precisely  similar  origin.  To  return  now  to  the  rings  seen  between  convex  glasses. 
Phenomenon  5.  If  homogeneous  light  be  used  to  illuminate  the  glasses,  the  rings  are  seen  in  much  greater 

VOL.  IV.  3  p 


466  LIGHT. 

Light       number,  and  the  more  according  to  the  degree  of  homogeneity  of  the  light.     When  this  is  as  perfect  as  possible,    Part  III. 
— s,--— '  as,  for  instance,  when  we  use  the  flame  of  a  spirit  lamp  with  a  salted  wick,  as  proposed  by  Mr.  Talbot,  they  are  v*— -V-— ' 
Phenomena  literally  innumerable,  extending  to  so  great  a  distance  that  they  become  too  close  to  each  other  to  be  counted,  or 
D*  honf  e    even  distinguished  by  the  naked  eye,  yet  still  distinct  on  using  a  magnifier,  but  requiring  a  higher  and   higher 
neous  lifht.   power  as  they  become  closer,  till  we  can  pursue  them  no  farther,  and  disappearing  from  their  closeness,  and  not 
from  any  confusion  or  running  of  one   into  the  other.     Moreover,  they  are  now  no  longer  composed  of  various 
colours,  but  are  wholly  of  the  colour  of  the  light  used  as  an   illumination,  being  mere  alternations  of  light  and 
obscurity,  and  the  intervals  between  them  being  absolutely  black. 

644.  Phenomenon  6.     When  the  illuminating  light  is  changed  from  one  homogeneous  ray  to  another,  as  when,  for 
Contraction  instance,  the  colours  of  the  prismatic  spectrum  are  thrown  in  succession  on  the  glasses  at  their  point  of  contact, 

the  rings  at  such  an  angle  as  to  be  reflected  to  the  eye,  then,  the  eye  remaining  at  rest,  the  rings  are  seen  to  dilate  and 

refraneibfe    contract  m  magnitude  as  the  illumination  shifts.     In  red  light  they  are  largest,  in  violet  least,  and  in  the  inter- 

,™  mediate  colours  of  intermediate  size.     Newton,  by  measuring  their  diameters,   ascertained  that  the   interval   of 

the  surfaces  or  thickness  of  the  plate  of  air,  where  the  violet  ring  of  any  order  was  seen,  is  to  its  thickness, 

where  the  corresponding  red  ring  of  the  same  order  is  formed,  nearly  as  9  :  14  ;  and,  determining  by  this  method, 

the  thickness  of  the  plate  of  air  where  the  brightest  part  of  the  first  ring  was  formed,  when  illuminated  in  suc- 

\nalysis  of    cession  by  the  several  rays  proceeding  from  the  extreme  red  to  the  extreme  violet,  he  ascertained  those  thick- 

,he  coloured  nesses  to  be  the  halves  of  the  numbers  already  set  down  in  the  second  column  of  the  Table,  p.  453,  expressed  in 

parts  of  an  inch,  and  which  answer  to  the  values  of — ,  or  the  lengths  of  a  semiundulation  for  each  ray. 

A 

645.  This  phenomenon  may  be  regarded  as  an  analysis  of  what  takes  place  when  the  rings  are  seen  in  white  light ; 
Synthesis  of  for  in  that  case  they  may  be  regarded  as  formed  by  the  superposition  one  on  the  other  of  sets  of  rings  of  all  the 
the  coloured  simple  colours,  each  set  having  its  own  peculiar  series  of  diameters.     The  manner  in  which  this  superposition 

lgs'  takes  place,  or  the  synthesis  of  the  several  orders  of  colours,  may  be  understood  by  reference  to  fig.  134,  where 

the  abscissae  or  horizontal  lines  represent  the  thicknesses  of  a  plate  of  air  between  two  glasses,  supposed  to 
increase  uniformly,  and  where  R  R',  RR",  &c.  represent  the  several  thicknesses  at  which  the  red,  in  the  system 
of  rings  illuminated  by  red  rings  only,  vanishes,  or  at  which  the  darkness  between  two  consecutive  red  rings  is 
observed  to  happen,  while  R  r,  Rr',  Rr",  &c.  represent  those  •  which  the  brightness  is  a  maximum.  In  like 
manner,  let  0  0',  0  0'',  &c.  be  taken  equal  to  the  several  thicknesses  at  which  the  orange  vanishes,  or  at  which 
the  black  intervals  in  the  system  of  orange  rings  are  seen,  and  so  on  for  the  yellow,  green,  blue,  indigo,  and 
violet  rings.  So  that  R  R',  0  0',  Y  Y',  &c.  are  to  each  other  in  the  ratio  of  the  numbers  in  column  2  of 
the  above  Table,  (Art.  575.)  Then  if  we  describe  a  set  of  undulating  curves  as  in  the  figure,  and  at 
any  point,  as  C  in  A  E,  draw  a  line  parallel  to  A  V,  cutting  all  these  curves  ;  their  several  ordinates,  or  the 
portions  of  this  line  intercepted  between  the  curves  and  their  abscissa?,  will  represent  the  intensity  of  the 
light  of  each  colour,  sent  to  the  eye  by  that  thickness  of  the  plate  of  air.  Hence,  the  colour  seen  at  that 
thickness  will  be  that  resulting  from  the  union  of  the  several  simple  rays  in  the  proportions  represented  by  their 
ordinates. 

g.jg  The  figure  being  laid  down  by  a  scale,  we  may  refer  to  it  to  identify  the  colours  of  particular  points.     Thus, 

Synthesis  of  nrst  at  the  thickness  0,  or  at  A  the  origin  of  the  tints,  all  the  ordinates  vanish,  and  this  point,  therefore,  is  black. 
the  several  As  the  thickness  of  the  plate  of  air  increases  from  0  while  yet  very  small,  it  is  evident,  on  inspection,  that  the 
orders  of  ordinates  of  the  several  curves  increase  with  unequal  rapidities,  those  for  the  more  refrangible  rays  more  rapidly 
than  those  for  the  less,  so  that  the  first  feeble  light  which  appears  at  a  very  small  thickness  A  1,  will  have  an 
excess  of  blue  rays,  constituting  the  pure  but  faint  blue  of  the  first  order,  (Art.  635.)  At  a  greater  thickness, 
however,  as  A  2,  the  common  ordinate  passes  nearly  through  the  maxima  of  all  the  curves,  being  a  little  short  of 
that  of  the  red,  and  a  little  beyond  that  of  the  violet.  The  difference,  however,  is  so  small,  that  the  several 
colours  will  all  be  present  nearly  in  the  proportions  to  constitute  whiteness,  and  being  all  nearly  at  their  maxi- 
mum, the  resulting  tint  will  be  a  brilliant  white.  This  agrees  with  observation ;  the  white  of  the  first  order 
being,  in  fact,  the  most  luminous  of  all ;  beyond  this  the  violet  falls  off  rapidly,  the  red  increases,  and  the  yellow 
is  nearly  at  its  maximum,  so  that  at  the  thickness  A  3  the  white  passes  into  yellow,  and  at  a  still  greater 
thickness,  A  4,  where  the  violet,  indigo,  blue,  and  green,  are  all  nearly  evanescent,  the  yellow  falling  otf,  and 
the  orange  and  red,  especially  the  latter,  in  considerable  abundance,  the  tint  resulting  will  be  a  fiery  orange, 
growing  more  and  more  ruddy.  At  B  is  the  minimum  of  the  yellow,  i.  e.  of  the  most  luminous  rays.  Here 
then  will  be  the  most  sombre  lint.  It  will  consist  of  very  little  either  of  orange,  green,  blue,  or  even  indigo ; 
but  a  moderate  portion  of  violet  and  a  little  red  will  produce  a  sombre  violet  purple,  which,  since  the  more  re- 
frangible rays  are  here  all  on  the  increase,  while  the  less  are  diminishing,  will  pass  rapidly  to  a  vivid  blue,  as  at 
the  thickness  denoted  by  A  5.  At  6,  where  the  ordinate  passes  through  the  maximum  of  the  yellow,  there  is 
almost  no  red,  very  little  orange,  a  good  deal  of  green,  very  little  blue,  and  hardly  any  indigo  or  violet  Here 
then  the  tint  will  be  yellow  verging  to  green,  but  the  green  is  diminishing  and  the  orange  increasing,  so  that  the 
yellow  rapidly  loses  its  green  tinge,  and  becomes  pure  and  lively.  At  7  the  predominant  rays  are  orange  and 
yellow,  being  so  copious  that  the  little  red  and  violet  with  which  they  are  mixed  does  not  prevent  the  tint  from 
being  a  rich,  high-coloured  yellow.  At  8  a  full  orange  and  copious  red  are  mixed  with  a  good  deal  of 
indigo  and  a  maximum  of  violet,  thus  producing  a  superb  crimson.  At  C  we  have  again  a  minimum  of 
yellow ;  but  there  being  at  the  same  time  a  maximum  of  red  and  indigo,  this  point,  though  dark  in  com- 
parison of  that  on  either  side,  will  still  be  characterised  by  a  fine  ruddy  purple.  This  completes,  and  as  we 
see  faithfully  represents,  the  second  order  of  colours.  At  9,  10  we  see  the  origin  of  the  vivid  green  of  the  third 
order,  in  the  comparative  copiousness  of  green,  yellow,  and  blue  rays  at  the  former  point,  and  of  yellow,  green, 


LIGHT.  467 

Light.      ftnd  violet  at  the  latter,  while  the  red  and  orange  are  almost  entirely  absent,  and  thus  we  may  pursue  all  the    Pan  III. 
-— y— J  tints  in  the  scale  enumerated  in  Art.  635  with  perfect  fidelity.  ^— ~^-^- 

As  the  thickness  increases,  however,  it  is  clear  that  rays  differing  but  little  in  refrangibility  will  differ  much  in      647. 
intensity,  as  the  smallest  difference  in  the  lengths  of  the  bases  of  their  curves  being  multiplied  by  the  number  of  Degradation 
times  they  are  repeated,  will  at  length  bring  about  a  complete  opposition,  so  that  the  maximum  of  one  ray  will of  thc  tmt!>' 
fall  at  length  on  a  minimum  of  another  differing  little  in  refrangibility,  and  not  at  all  in  colour.     Thus,  at  con- 
siderable thicknesses,  such  as  the  10th  or  20th  order,  there  will  coexist  both  maxima  and  minima  of  every  colour ; 
since  each  colour,  in  fact,  consists  not  of  rays  of  one  definite  refrangibility,  but  of  all  gradations  of  refrangibility 
between  certain  limits.     In  consequence,  the  tints,  as  the  thickness  increases,  will  grow  less  and  less  pure,  and 
will  at  length  merge  into  undistinguishable  whiteness,  which,  however,  for  this  very  reason,  will  be  only  half  as 
brilliant  as  the  white  of  the  first  order,  which  contains  all  the  rays  at  their  maximum  of  intensity. 

Phenomenon  7.     Such  are  the  phenomena  when  a  plate  of  air  is  included  between  two  surfaces  of  glass.     It  is      648 
not  however  as  air,  but  as  distance,  that  it  acts ;  for  in  the  vacuum  of  an  air-pump  the  rings  are  seen  without  Colours 
any  sensible  alteration.     If,  however,  a  much  more  refracting  mediunt,  as  water  or  oil,  be  interposed,  the  dia-  "^j1' 
meters  of  the  rings  are  observed  to  contract,  preserving,  however,  the  same  colours  and  the  same  laws  of  their  Sig-^ent 
breadths  ;  and  Newton  found  by  exact  measurements,  that  the  thicknesses  of  different  media  interposed,  at  which  media. 
a  given  tint  is  seen,  are  in  the  inverse  ratio  of  their  refractive  indices.     Thus,  the  white  of  the  first  order  being 

produced  in  vacuo  or  air  at  the  178000th  of  an  inch,  will  be  produced  in  water  at  —  — —  pirt  of  that  thickness. 

1  *3oo 

He  found,  moreover,  that  the  law  stated  in  Art.  639  for  the  dilatation  of  the  rings  by  oblique  incidence,  holds 
equally  good,  whatever  be  the  nature  of  the  interposed  medium.  Hence  it  follows,  that  in  dense  media  the 
dilatation  at  great  obliquities  is  much  less  than  in  rare  ones,  and  that  in  consequence  a  given  thickness  will  re- 
flect a  colour  much  less  variable  by  change  of  obliquity  when  the  medium  has  a  high  refractive  power  than  when 
low.  Thus,  the  colours  of  a  soap  bubble  vary  much  less  by  change  of  incidence  than  those  of  a  film  of  air,  and 
much  more,  on  the  other  hand,  than  the  iridescent  colours  on  polished  steel,  which  arise  from  a  film  of  oxide 
formed  on  the  heated  surface. 

Phenomenon  8.     Surfaces  of  glass,  or  other  denser  medium  enclosing  the  thin  plate  of  a  rarer,  are  not  how-      649. 
ever  necessary  to  the  production  of  the  colours  ;  they  are  equally,  and  indeed  more  brilliantly,  visible  when  any  Colours  "e- 
very  thin  laminae  of  a  denser  medium  is  enclosed  in  a  rarer,  as  in  air,  or  in  vacuo.     Thus,  soap  bubbles,  exceed-  flecleii  by 
ingly  thin  films  of  mica,  &c.  exhibit  the  same  succession  of  colours,  arranged  in  fringes  according  to  the  variable  f?ap  I", 
thickness  of  the  plates.     The  following  very  beautiful  and  satisfactory  mode  of  exhibiting  the  fringes  formed  by 
plates  of  glass  of  a  tangible  thickness  has  been  imagined  by  Mr.  Talbot.     If  a  bubble  of  glass  be  blown  so  thin 
as  to  burst,  and  the  glass  films  which  result  be  viewed  in  a  dark  room  by  the  light  of  a  spirit  lamp  with  a  salted 
wick,  they  will  be  seen  to  be  completely  covered  with  stria,  alternately  bright  and  black,  in  undulating  curves 
parallel  to  each  other  according  to  the  varying  thickness  of  the  film.     Where  the  thickness  is  tolerably  uniform, 
the  stria;  are   broad  ;  where  it  varies  rapidly,  tlit-y   become  so  crowded  as  to  elude  the  unassisted   sight,  and 
require  a  microscope  to  be  discerned.     If  the  film  of  glass  producing  these  fringes  be  supposed  equal  to   the 
thousandth  of  an  inch  in  thickness,  they  must  correspond  to  about  the  89th  order  of  the  rings,  and  thus  serve  to 
demonstrate  the  high  degree  of  homogeneity  of  the  light ;   for  if  the  slightest  difference  of  refrangibility  existed, 
its  effect  multiplied  eighty-nine  times  would  become  perceptible  in  a  confusion  and  partial  obliteration  of  the 
black  intervals.     In  fact,  the  thickness  of  a  plate  at  which  alternations  of  light  and  darkness  or  of  colour  can 
no  longer  be  discerned,  is  the  best  criterion  of  the  degree  of  homogeneity  of  any  proposed  light,  and  is,  in  fact, 
a  numerical  measure  of  it.     This  experiment  is  otherwise  instructive,  as  it  shows  that  the  property  of  light  on 
which  the  fringes  depend  is  not  restricted  to  extremely  minute  thicknesses,  but  subsists  while  the  light  traverses 
what  may  be  comparatively  termed  considerable  intervals. 

Phenomenon  9.     When  the  glasses  between  which  the  reflected  rings  are  formed  are  held  up  against  the  light,       650. 
a  set  of  transmitted  coloured  rings  is  seen,  much  fainter,  however,  than  the  reflected  ones,  but  consisting  of  tints  Transmitted 
complementary  to  those  of  the  latter,  i.  e.  such  as  united  with  them  would  produce  white.     Thus  the  centre  is  colours- 
white,  which  is  succeeded  bv  a  yellowish  tinge,  passing  into  obscurity,  or  black,  which  is  followed  by  violet  and 
blue.     This  completes  the  series  of  the  first  order.     Those   of  the   second  are  white,  yellow,  red,  violet,  blue  : 
of  the  third,  green,  yellow,  red,  bluish  green,  after  which  succeed  faint  alternations  of  red  and  bluish  green,  the 
degradation  of  tints  being  much  more  rapid  than  in  the  reflected  rings. 

It  was  to  explain  these  phenomena  that  Newton  devised  his  doctrine  of  the  fits  of  easy  reflexion  and  trans-       651. 
mission,  mentioned  in  the  9th  postulate  of  Art.  526.    This  doctrine  we  shall  now  proceed  to  develope  further,  and  Newton's 
apply,  as  he  has  done,  to  the  case  in  question.     In  addition  then  to  the  general  hypothesis  there  assumed,  it  will  e*Planat>os 
be  necessary  to  assume  as  follows  :  of. the 

The  intervals  at  which  the  fits  recur,  differ  in  different  rays  according  to  their  refrangibilities,  being  greatest  for  thin  ^ates. 
the  red  and  least  for  violet  rays,  and  for  these,  and  the  intermediate  rays,  in  vacuo,  and  at  a  perpendicular  inci-      652. 
dence,  are  represented  in  fractions  of  an  inch  by  the  halves  of  the  numbers  in  column  2  of  the  Table,  Art.  575.    Laws  of 

In  other  media,  the  lengths  of  the  intervals  in  the  course  of  a  molecule  at  which  its  fits  recur  are  shorter,  in  tne  fits- 
the  ratio  of  the  index  of  refraction  of  the  medium  to  unity.  653. 

At  oblique  incidences,  or  when  a  ray  traverses  a  medium  after  being  intromitted  obliquely,  (at  an  angle  =  0      gt< 
with  the  internal  perpendicular,)  the  lengths  of  the  fits  are  greater  than  at  a  perpendicular  incidence,  in  the 
ratio  of  radius  to  the   rectangle  between  the  cosine  of  0  and  the  cosine  of  an  arc  u,  given   by   the   equation 
106/.+  1    . 

Sin"=-To7>-Sm*- 

3r2 


468  LIGHT. 

Light.          Let  us  now  consider  what  will  happen  to  a  luminous  molecule,  the  length  of  whose  fits  in  any  medium  is  \  X,    Part  III 
>-— - V"^1  which,  having  been  intromitted  perpendicularly  at  the  first  surface,  and  traversing  its  thickness  (=  t),  reaches  the  ^.^-^^-^ 

655.  second.     First,  then,  if  we  suppose  t  an  exact  multiple  of  J  X,  it  is  evident  that  the  molecule  will  arrive  at  the 
Explanation  second  surface  in  precisely  the  same  phase  of  its  fit  of  transmission  as   at  the  first.     Of  course  it  is  placed  in 

Ss  the  very  same  circumstances  in  every  respect,  and  having  been  transmitted  before  must  necessarily  be  so  again. 
mo"encour  Thus  every  ray  which  enters  perpendicularly  into  such  a  lamina  must  pass  through  it,  and  cannot  be  reflected  at 
light.  its  second  surface.  On  the  other  hand,  if  the  thickness  of  the  lamina  be  supposed  an  exact  odd  multiple  of 

J  X,  &c.  every  molecule  intromitted  at  its  first  surface  will  on  its  arrival  at  the  second  be  in  exactly  the  contrary 
phase  of  its  fits,  and,  having  been  before  in  some  phase  of  a  fit  of  transmission,  will  now  be  in  a  similar  phase  of 
a  fit  of  reflexion.  It  will,  therefore,  not  necessarily  be  transmitted ;  but  a  reflexion,  more  or  less  copious,  will 
take  place  at  the  second  surface  in  this  case,  according  to  the  nature  of  the  medium  and  its  general  action  on 
light.  For  it  will  be  remembered,  that  every  molecule  in  a  fit  of  reflexion  is  not  necessarily  reflected.  It  is 
disposed  to  be  so ;  but  whether  it  will  or  no,  will  depend  on  the  medium  it  moves  in  and  that  on  which  it 
impinges,  and  on  the  phase  of  its  fit.  Now  conceive  an  eye  placed  at  a  distance  from  a  lamina  of  unequal 
thickness,  so  as  to  receive  rays  reflected  at  a  very  nearly  perpendicular  incidence  from  it.  It  is  evident,  that  in 
virtue  of  the  reflexion  from  the  first  surface,  which  is  uniform,  it  will  receive  equal  quantities  of  light  from  every 
point.  But  with  regard  to  the  light  reflected  from  the  second  the  case  is  different ;  for  in  all  those  parts  where 
the  thickness  of  the  lamina  is  an  exact  even  multiple  of  |  X,  none  will  be  reflected,  while  in  all  those  where  it  is 

an  exact  odd  multiple  of  — ,  a  reflexion  will  take  place  ;  and  since  each  molecule  so  reflected  retraces  the  path 
4 

by  which  it  arrived,  and  therefore   describes  again  the  same  multiple  of  — ;  its  total  path  described  within   the 

lamina,  when  it  has  reached  the  first  surface  again,  will  be  an  exact  multiple  of  — ,  and  therefore  it  will  pene- 
trate that  surface  and  reach  the  eye.  In  consequence,  in  virtue  of  the  reflexion  at  the  second  surface  alone,  thf. 
lamina  would  appear  black  in  every  part  where  its  thickness  =  0,  or  -j->or  — ,  &c.,  and  bright  in  those  parts 

where  its  thickness  =   — — ,  or  — -,  '—- ,  &c.  ad  hiftniliim.     In  the  intermediate  thicknesses  it  would  have  a 
4  44 

brightness  intermediate  between  these  and  absolute  obscurity  ;  so  that  on  the  whole,  the  lamina  would  appear 
marked  all  over  with  dark  and  bright  alternating  fringes,  just  as  we  see  it  actually  does  in  the  experiment 
described,  (Art.  649.)  The  uniform  reflexion  from  the  first  surface  superposed  on  these,  will  not  prevent  their 
inequality  of  illumination  from  being  distinctly  seen. 

656.  Hence  it  is  evident,  that  if  we  take  the  abscissa  of  a  curve  equal  to  thickness  of  the  lamina  at  any  point,  and 
Oi  the         the  ordinate  proportional  to  the  intensity  of  the  light  reflected  from  the  second  surface,  and  returned  through  the 
rings  seen     first,  this  curve  will  be  an  undulating  line,  such  as  we  have  represented  in  fig.  134,  touching  the  abscissa  at  equal 
by  white      distances  equal  to  the  length  of  a  whole  fit  of  a  ray  of  the  colour  in  question.     Now  these  distances  for  rays  of 

different  colours  being  supposed  such  as  we  have  assumed  in  Art.  652,  the  construction  of  Art.  645  holds 
good,  and  when  white  light  falls  on  the  lamina,  its  second  surface  will  reflect  a  series  of  colours  of  the  composi- 
tion there  demonstrated,  and  such  as  we  actually  observe,  but  diluted  with  the  light  uniformly  reflected  from 
every  point  of  the  first  surface. 

If  the  lamina  instead  of  a  vacuum  be  composed  of  any  refracting  medium,  the  tints  will  manifestly  succeed 
each  other  in  a  similar  series,  but  the  thickness  at  which  they  are  produced  will  be  to  that  in  a  lamina  of  vacuum, 
in  the  ratio  of  the  lengths  of  the  fits  in  the  two  cases,  that  is,  in  the  proportion  of  i  :  the  index  of  refraction  of 
the  medium.  Thus  the  rings  seen  between  two  object  glasses  including  air,  ought  to  contract  when  water,  oil, 
&c.  is  admitted  between  them,  as  they  are  found  to  do,  and,  by  measure,  iu  that  precise  ratio. 

657.  At  oblique  incidences,  0  being  the  angle  of  intromission  into  the  lamina,  t  .  sec  0  is  the  whole  path  of  the  ray 
Of  thedila-  between  the  first  and  second  surfaces,  and  since  £  X  .  sec  0  .  sec  u  is  the  length  of  the  fits  of  the  given  ray  at 
tatioQof  the  this  obliquity,  in   order  that  the  luminous   molecule  may   arrive  at  the  second  surface  in  the  same  phase,  and 
rings  at        therefore  be  reflected  with  equal  intensity,  it  must  in  this  space  have  passed  over  the  same  number  of  these  fits  ; 

oblique 

o  /      ^cc  v 

incidences.   nence  we  must  have =  constant,  or  t  proportional  to  sec  u,  which  agrees  with  observation 

X  .  sec  0  .  sec  u 

65g  All  the  light  which  is  not  reflected  at  the  second  surface  passes  through  it,  and  forms  the  transmitted  series  of 

Of  the         colours.     These,  therefore,  consist  of  the  whole  incident  light  (=  1)   minus  that  reflected  at  the  first  surface, 

transmitted  (which  will  be  a  small  fraction,  and  which  we  will  call  a,)  minvs  that  reflected  at  the  second  surface.     Now  this 

"nss  last  will   be  a  periodical  function  whose  minimum  is  0,  and  its   maximum   can   never  exceed   a,  because  the 

reflexion  at  the  second  surface  of  a  medium   cannot  be  stronger  than  at  the  first  at  a  perpendicular  incidence. 

We  may  then  represent  it  by  a  (sin  —  I ,  and  thus  we  have  I  —  «  4  1  -|- 1  sin  —  V  j-  for  the  intensity  of  this 

/      2  l\ 

particular  coloured  ray  in  the  transmitted  series,  and  a  f  sin —  I    in   the    reflected.      Hence  it   is  evident,    that 

owing  to  the  smallness  of  a,  the  difference  between  the  brightest  and  darkest  part  of  the  transmitted  series  will 
be  small  in  comparison  with  the  whole  light,  and  thus  the  alternations  in  homogeneous  light  ought  to  be  (as 
they  are)  much  less  sensible  than  in  the  reflected  rings,  and  the  tints  in  white  light  much  more  pallid  and  dilute. 


LIGHT.  469 

Lignt.          Thus  we  see  that  the  Newtonian  hypothesis  of  the  fits  affords  a  satisfactory-enough  explanation,  or  rather    Part  III. 

••v"" ~^  represents  with  exactness  all  the  phenomena  above  described.     It  has  been  even  asserted,  that  this  doctrine  is  •— -^—— 

really  not  an  hypothesis,  but  nothing  moie  than  a  pure  statement  of  facts;   for  that,  first,  in  point  of  mere  fact,       659. 

the  second  surface  of  the  lamina  does  send  light  to  the  eye,  in  the  bright  parts  of  the  fringes,  and  does  not  send 

it  in  the  dark  parts  ;  and,  secondly,  that  this  is  the  same  thing  with  saying  that  the  light  which  has  traversed  a 

thickness  =  (2  n  -f-  1)  —  is,  and  that  which  has  traversed  2  n  — -  is  not  susceptible  of  being  reflected.    And, 

in  truth,  if  only  one  ray  could  be  regarded  as  being  concerned,  and  were  the  light  reflected  at  the  first  surface 
of  the  lamina  altogether  out  of  the  question,  this  way  of  stating  it  would  be  strictly  correct.  But,  if  it  can  be 
shown,  that,  on  any  other  hypothesis  of  the  nature  of  light,  (as  the  undulatory,)  the  second  link  of  this  argument 
is  invalid ;  and  that  though  the  second  surface,  like  the  first,  may  reflect  in  every  part,  without  regard  to  its 
thickness,  its  full  average  portion  of  the  light  that  is  incident  on  it ;  yet  that  afterwards,  by  reason  of  the 
interference  of  rays  reflected  from  the  first  surface,  such  light  does  not  reach  the  eye  (being  destroyed  in  every 

point  of  its  course)  from  those  parts  where  the  thickness  is  an  even  multiple  of  — — ,  then  it  is  evident,  that  the 

Newtonian  doctrine  is  something  more  than  a  mere  alittr  statement  of  facts,  and  is  open  to  examination  as  a 
theory. 

Let  us  now  see,  therefore,  what  account  the  undulatory  theory  gives  of  these   phenomena.     We  will  begin,      gfio 
for  a  reason  which  will  presently  appear,  with  the  transmitted  rings.     Conceive,  then,  a  ray,  the  length  of  Explanation 
whose  undulations  in  any  medium  is  X,  to  be  incident  perpendicularly  on  the  first  surface  of  a  lamina  of  that  of  the 
medium  whose  thickness  is  —  t;   and  (for  simplicity)  let  its  surfaces  be  supposed  parallel,  then  it  will  be  transmitte^ 
divided  into  two  portions,  the  first  (  =  a)  reflected,  and  the  second  (=  1  -a)  intromitted.     Let  0  be  the  phase  "nduhJorv" 
of  this  portion  at  reaching  the  second  surface.     Here  it  will  be  again  divided  into  two  portions,  the  one  hypothesis. 
reflected  back  into  the  medium  and  equal  to  (1  —  a)  .  a,  or  (a  being  small)  very  nearly  to  a,  and  the  remainder 
(1  —  a)  —  a  (1  —  a),  or  nearly  1  —  2  a,  transmitted.     These  portions,  supposing  no  undulation,  or  part  of  an 
undulation,  gained  or  lost  in  the  act  of  transmission  or  reflexion,  will  both  be  in  the  phase  0.     The   reflected 

t 

portion  will  again  encounter  the  first  surface  in  the  phase  0  -f-  2  IT  .  — ,  will  there  be  again  partially  reflected, 

A. 

wUh  an  intensity  equal  to  a  x  a  —  a1,  and  the  portion  so  reflected  will  reach  the  second  surface  in  the  phase 
0  -j-  2  TT  .  -  — ,  and  will  there  be  transmitted  with  an  intensity  =  (1  —  a)  .  a8,  or  nearly  =r  a*.  Now,  the 

A 

reflexions  being   all   perpendicular,  this   portion  will    be  confounded   with   the   portion   1    —  2  a  transmitted 

without  any  reflexion ;  and  putting  "  =  "^\  —  •£  a  =  1  —  a  nearly,  and  a'  =  */a*  =  a,  a  and  a'  will  represent 
the  amplitudes  of  vibration  of  the  ethereal  molecule  at  the  posterior  surface,  which  each  of  these  rays  tend  to 
impress  on  it.  Hence,  its  total  excursion  from  rest  will  be  represented  by 


that  is 


/  2  t  \ 

a  .  COS  0  +  a' .  COS  I  0  -f-  2  TT  .  — -  J, 

/  "  t  \ 

(1  -  a)  cos"  -j-  a  .  cos  I  0  -f  2  TT  .  — —  J. 

/  2  t\ 

=  1  cos  0  -f-  a  .  cos  I  0  -\-  2  v  . J  -  a  .  cos  0. 


The  first  term  of  this  is  independent  of  t,  and  represents,  in  fact,  the  incident  ray  in  the  state  in  which  it  would 
arrive  at  the  second  surface,  had  no  reflexions  taken  place.     The  other  two  terms  represent  rays  the  former  of 

which  evidently  is  in  complete  discordance  with  the  latter,  and  destroys  it  when  t  is  any  odd  multiple  of  — ,  (or  of 

the  half  length  of  one  of  Newton's  fits,  a  fit  being,  as  we  have  seen  above,  equal  to  half  an  undulation,)  thus 
leaving  the  ray  at  its  emergence  of  the  same  intensity  as  it  would  have  had  were  the  lamina  away  ;    but  when  t 

is   any  odd  multiple  of  half  a  fit,  then  the  value  of  cos  (  0  -f  2  v  .  -  -  J  =  -  cos  0 ;  and  the  emergent  ray 

is  in  this  case  represented  by  (I  -  -2  a)  .  cos  0,  being  less  than  the  incident  ray  by  twice  the  light  reflected  at 
the  first  surface. 

Thus  if  the   thickness  of  the   plate  be  different  in   different    parts,   the   light    transmitted   through    it   to      661 
the  eye  will  not  be  uniform,  but  will  have  alternate  maxima  and  minima  corresponding  to  the  thicknesses  0  Origin  of 
\  2  X  3  X  the'bright 

-j-»  r- »        —7—,    &C.  and  dark 

444  u 

rings  in  no- 
li we  apply  to  the  expression  above  given,  the  general  formula  Art    (613)  for  the  composition  of  rays  in  one  J!"¥ene°l>s 
plane,  we  shall  find  for  the  intensity  A4  of  the  ray  finally  emergent. 


Light. 


663. 

Transmit- 
ted tints  in 
white  light 
expressed 
alge- 
braically. 


470  LIGHT. 

A»  =  (I  -  «)»  +  2  a  (1  -  a)  .  cos  2  «•  .  -^-  +  o!  ' 

(<  V 
2   T    — J 

(<    V 
2    IT  —    \ 

which  shows  that  the  several  maxima  are  equal  to  the  incident  ray,  and  the  minima  to  that  ray  diminished  by 
four  times  the  light  reflected  at  the  first  surface.  The  difference  of  phase  between  the  simple  and  composite 
emergent  ray,  or  the  value  of  B  in  the  formula  cited,  is  given  by  the  equation, 

a  /  2 1  \  /  2  t  \ 

sin  B  =  -T-  .  sin  (  2  •*• .  — —  I  =  a  .  sin  I  2  v  ,  — —  I,  neglecting  As, 
A  \  X  /  \  X  / 

so  that  for  such  media  as  have  not  a  very  high  refractive  power,  this  difference  is  always  small.  It  is,  however, 
periodical,  and  differs  for  different  thicknesses. 

Suppose  now,  instead  of  homogeneous  light,  white  light  to  fall  on  the  lamina,  and  let  us  represent  a  ray  of 
such  light,  as  in  Art.  488,  by  C  +  C'  +  C"+  &c.,  or  by  S  (C) ,  C,  C',  &c.  being  the  intensity  of  the  several 
elementary  rays  of  all  degrees  of  refrangibility,  then  will  the  transmitted  compound  beam  be  represented  in  tint 
and  intensity  by 


Part  III. 


or  by 

Now  this  is  the  same  with 


CJ1  -4«.sin^27r  -LY  j-j-c'  Jl  -  4«.  sin  ^2  v  .  -^j   j  +  &c. 
S.  C  |l  -  4  a.  sin  f2  *•  —  J    >. 

S  JC  (1  -  4  a)  +  C  (4  a  -  4  a  .  sin  (  2  «•  .  -—  -J   j  = 
=  (1  -  4  a)  .  S  (C)  +  4  a  .  S   j  C  .  cos  f  2  v  .  -jj-J  ]  • 


The  first  term  of  this  expression  represents  a  beam  of  white  light  of  the  intensity  1  -  4  a.  The  second,  a 
compound  tint  of  the  intensity  4  a,  which,  diluted  with  the  above-mentioned  white  light,  forms  the  pallid  tints 
of  the  transmitted  series.  If  we  disregard  this  dilution,  and  con&ider  only  the  tint  in  its  purity  as  it  would 
appear  were  the  white  light  suppressed,  its  expression 

4o.S  {c.cos(2*-.  -liY  |  =  4o{s(C)  -  S  (c  .  sin  (2  r  .  -^  J)} 
indicates  that  it  is  complementary  to  the  tint  represented  by 


/  2  t  V 

But  if  we  conceive  a  curve  whose  abscissa  =  t,  and  whose  ordinate  is  C  .  sin  (  2  v  .  — — -  J ,  it  is  evident  that 

this  will  be  precisely  the  undulating  curve  represented  for  each  prismatic  ray  in  fig.  134 ;  and  taking  the  sum  of 
all  the  ordinates  so  drawn  for  each  colour  in  the  spectrum,  we  have  the  identical  construction  from  which  we 
derived  the  colours  of  the  reflected  rings  in  Art.  645.  If,  then,  we  take  the  series  of  tints  so  composed,  and 
thence  deduce  their  complements  to  white  light,  and  dilute  these  complementary  colours  with  white,  in  the 
proportion  of  4  a  rays  of  the  complementary  colour  to  1  —  4  a  of  white,  we  shall  have  the  series  of  transmitted 
tints  which  ought  to  result  from  the  doctrine  of  interferences,  and  which,  in  fact,  is  observed. 

664.          In  the  case  of   oblique  transmission,  let  AC,  B  D,  fig.  135,  be  the  surfaces  of  the  lamini,   and  A  a  its 

Case  of        thickness  ;  and  let  A  E  be  the  surface  of  a  wave  of  which  the  point  A  has  just  reached  the  first  surface  of  the 

oblique        lamina  ;  and  let  S  A,  S  C,  perpendicular  to  it,  represent  rays  emanating  from  one  origin  S,  then  will  a  partial 

transmission  reflex;on  take  place,  and  its  intensity  will  be  diminished  in  some  certain  ratio  1  :  1  —  a  depending  on  the  angle 

Fig.  135.      op  mc;(jence>     The  transmitted  wave  will  be  bent  aside,  taking  the  position  A  b,  and  advancing  along  A  B  the 

refracted  ray ;  so  that  when  it  reaches  the  position  B  F,  the  wave  without  the  lamina  will  have  the  corresponding 

position  FG.     Here  another  partial  reflexion  will  take  place  depending  on  the  interior  incidence,  and  we  may 

denote  by  (1  —  a)  (1  —  a)  the  transmitted  portion,  and  by  (1  —  a)  .  a  the  reflected  portion.     These  portions  set 

off  together,  from  B,  the  former,  with  the  velocity  V  due  to  the  exterior  medium,  along  the  line  B  H  parallel  to 

S  A,  forming  a  wave  which  (provided  S  be  sufficiently  distant)  may  be  regarded  as  a  plane  of  indefinite  extent 

moving  uniformly  with  that  velocity  along  B  H.     The  latter  portion  proceeds  along  B  C,  according  to  the  law 

of  reflexion,  with  the  velocity  V  due  to  the  medium  of  which  the  lamina  is  composed  till  it  reaches  C,  where  it 

undergoes  another  partial  reflexion,  and  proceeds  back  along  the  line  C  D  with  a  diminished  intensity  =  (1  —  a) 


LIGHT.  471 

1  jght.     .  a«,  but  with  the  same  velocity  V  till  it  reaches  D,  having  described  a  space  =  BC-f-CD  =  2AB  with  that     Part  III. 
—  v—  '  velocity.     At  D  it  undergoes  another  partial  reflexion,  and  only  a  portion  =  (1—  a)  (1  —  a)  .  as  is  transmitted,  ^—  ~v—  ' 
which  sets  off  from  D  along  the  line  D  I  (parallel  to  B  H)  with  the  velocity  V,  that  is,  with  the  same  velocity 
as  the  wave  along  B  H.     This  wave  may  also  be  regarded  as  a  plane  of  indefinite  extent  perpendicular  to  D  1, 
and  therefore  parallel  to  the  former.     But  they  are  not  coincident  ;   for  the  former,  having  the  start  of  the  latter, 
will  have  come  into  a  position  I  H  K  in  advance  of  the  position  D  L  M  taken  by  the  latter,  and  both  the  waves 
moving  forwards  now  with  the  same  velocity  V  will  preserve  this  distance  for  ever  unaltered.     The  interval  L  H 
we  may  term  the  interval  of  retardation.     To  determine  it,  we  have  to  consider  that  the  space  B  H  is  described 
by  the  former  wave  with  a  velocity  V,  while  the  latter  describes  B  C  -$-C  D  with  the  velocity  V,  and  therefore 

CD).-^-  =  2AB.^-=2<.  sec  ,>./*, 

putting  in  for  the  relative  index  of  refraction  of  the  lamina,  p  for  the  angle  of  refraction  a  A  B,  and  t  for  the 
thickness  A  a,  because  V  :  V  '.  '.  p  :  1. 

Again,  B  L  =  B  D  .  cos  DBL=  D  B  .  sin  0  (0  being  the  angle  of  incidence  corresponding  to  p  the  angle  of 
refraction,)  =  2  a  B  .  sin  0  =  2  t  .  tan  p  .  sin  0.     Therefore  the  whole  interval  of  retardation  is  equal  to 

2  t  .  p 
2  t  {  /t  .  sec  p  —  tan  p  .  sin  0  }  =  -  •  (I  —  sin  />*)  =  2  /»  t  .  cos  p 

COS  p 

because  sin  0  =  /»  .  sin  p. 

Thus,  in  virtue  of  the  two  internal  reflexions,  each  wave  which  before  entering  the  medium  was   single,  will      555 
after  quitting  it  be  double,  being  followed  at  the  constant  interval  2  /t  t  .  cos  p  by  a  feebler  wave  of  the  intensity 
above  determined.     The  same  being  true  of  every  wave  of  the  system  of  which  the  ray  consists,  these  two 
systems  (considered  as  of  indefinite  duration)  will  be  superposed  on,  and  interfere|with  each  other,  according  to 
the  general  principles  before  laid  down. 

Let  \  be  the  length  of  an  undulation  in  the  lamina,  then  will  /t  X  represent  that  of  an  undulation  in  the  sur-       666. 
rounding  medium.     This  is  obvious,  because  the  velocity  in  the  latter  being  to  that  in  the  former  as  fi  :  1  ;    and  Undulatiom 
the  same  number  of  undulations  being  propagated  in  the  same  time  through  a  given  point  in  both  cases,  they  s'lorter  in 

must  be  more    crowded,    and   therefore  occupy  less  space  in  the  one  than   the  other  in  the  ratio   of  the  denf.er 

i      •,•  medis. 

velocities. 

Hence  the  differences  of  phases  between  -;he  interfering  systems  at  any  point  will  equal  fifi7 

interval  of  retardation  2  t  .  cos  p  2  I1  Genera! 

•2*.-  —  =  2*-.-  -   =2*-.--—,   putting  tr=t.COSp,  express.on 

p,  \  XX  for  the 

transmitted 
and  theretore  the  final  resulting  wave  will  be  expressed  by  the  equation  ray 


X  =  */  (\  -  a)  (1^-  «)  I  cos  0  -f  a  .  cos  (o  -f  2  w  .  ^-\  \  , 
which  being  resolved  in'o  the  fundamental  form  A  .  cos  (0  -f-  B),  as  before,  gives 

A«  =  (1  -  a)  (1  -  a)  .  I  1  +  2  a  .  cos  (2  TT  .   ^\  -f- 


and 

sin  B  = 


1  -f  2  a  .  cos      2  IT  .  ~      +  a 


(2  IT  .  ~\ 


Such  are  the  general  expressions  for  the  intensity  and  change  of  origin  of  the  compound  transmitted  ray.  568 
It  is  evident,  however,  that  when  a  and  a.  are  small,  which  they  always  necessarily  are  in  any  but  extreme  cases  Case  of 
this  value  of  A*  reduces  itself  by  neglecting  their  powers  and  products  to  moderate 

obliuuititf. 

/  41    \« 

(1  —  a  -f-  a)  —  4  o  .  si 

which  is  exactly  analogous  to  the  expression  in  Art.  662,  for  the  case  of  perpendicular  incidence ;  and  shows, 
that  with  the  exception  of  a  very  trifling  difference  in  the  degree  of  dilution,  the  same  laws  of  alternation 
in  brightness,  in  homogeneous  light,  and  of  tint  in  white  light,  must  hold  good  in  both  cases. 

But  there  is  one  essential  difference.      The  same  tints  will  arise  in  the  case  of  oblique  incidence   at  the        .--Q 
thickness  t,  which  in  that  of  perpendicular  incidence  is  produced  at  the  thickness  t .  cos  p,  because^  =  t.  cos  p.  iyi 
Now  this  is  always  less  than  t,  and  therefore  the  tint  produced   at  oblique  incidences  at  the  given  thickness  of'the'rings 
will  be  higher  in  the  scale  (or  correspond  to  a  less  thickness)  than  in  perpendicular ;    and,  consequently,  the  explained, 
rings,  or  fringes,  so  seen  by  transmission  should  dilate  by  inclining  the  lamina  to  the  eye.     The  law  of  dilata- 
tion evidently,  at  moderate  incidences,  coincides  nearly  with  Newton  s  rule;    for  this  gives,  on  reduction. 
neglecting  sin  />*, 


472  LIGHT. 

Light  (  1  106    ,  I 

^_« ^s—'  sec  «  =  sec  />  •!  1  —   -   -    .    -—(/•  —  0  •  tan  />*    , 

(  2          107  J 

which  does  not  deviate  very  greatly  from  sec  p  at  moderate  incidences. 

670.  At  great  incidences  the  case  is  different,  and  the  noncoincidence  of  the  results  of  the  undulatory  doctrine 
Deviation  with  experiment  might  be  drawn  into  an  argument  against  it,  were  we  sure  that  the  law  of  refraction  at  extreme 
from  New-  ;ncidences,  and  wjtn  very  thin  laminae,  does  not  vary  sensibly  from  that  of  the  proportional  sines.  This  is, 
«°"at  obU  '"deed,  highly  probable,  as  M.  Fresnel  has  remarked,  (Mem.  «/r  la  Diffraction,  fyc.)  and  as  we  have  before 
qulties  pro-  had  occasion  to  observe.  The  inquiry  into  which  this  would  lead,  is,  however,  one  of  the  most  delicate  and  difficult 
bably  ac-  in  physical  optics,  and  the  reader  must  be  content  with  this  general  notice  of  a  possible  explanation  of  one  of  the 
rounted  for.  many  difficulties  which  still  beset  the  undulatory  doctrine. 

671  The  origin  of  the  reflected  rings  may  be  accounted  for  in  a  similar  way  from  the  partial  transmission  of  the 
Origin  of      waves  reflected  from  the  second  surface  back  through  the  first,  and  their  interference  with  the  waves  reflected 
the  reflected  immediately  from  the  first.   The  relative  intensities  of  these  waves,  (in  general,)  are  a  and  (1  —  a)  (1  -  a)  .  a  ; 
rings.           or>  m  tne  case  wnere  a  and  a  are  both  small,  nearly  in  the  ratio  of  a  :  a,  and  at  a  perpendicular  incidence,  very 

nearly  in  the  ratio  of  equality.  Hence  their  mutual  destruction  in  the  case  of  complete  discordance  will  be  much 
more  complete  than  in  the  transmitted  rings,  and  the  colours  arising,  much  less  dilute  than  those  of  the  latter, 
agreeably  to  observation. 

672  There  is,  however,  one  consideration  of  importance  to  be  attended  to  in  the  application  of  the  undulatory  doc- 
1.0SS  of  half  trine  to  the  reflected  rings,  which  at  first  sight  appears  in  the  light  of  a  powerful  argument  against  its  admis- 
an  undula-    sibility,  viz.  that  if  we  apply  the  same  reasoning  to  the  reflected,  as  we  have  already  done  to  the  transmitted, 
tion.             rings,  we  should  arrive  at  the  conclusion,  that  their  tints   should  be  precisely  the  same  and  in  the  same  order, 

beginning  with  a  bright  white  in  the  centre  ;  because  here,  the  path  traversed  by  the  ray  within  the  lamina 
vanishing,  the  waves  reflected  from  the  two  surfaces  ought  to  be  in  exact  accordance,  whereas  it  appears,  by 
observation,  that  the  reverse  is  the  case,  the  central  spot  being  black  instead  of  white.  It  becomes  necessary, 
then,  to  suppose,  that  in  this  else,  half  an  undulation  is  lost  or  gained  either  by  the  wave  reflected  from  the  first 
or  second  surface.  If  this  hypothesis  be  made,  the  phenomena  of  the  reflected  rings  are  completely  represented 
on  the  undulatory  system,  for  the  compound  wave  reflected  by  the  joint  action  of  the  two  surfaces  should  be 
represented  by  the  equation, 

X  =  VT.  cos  0 -)-  v'a(l-a)  (I  -«).  cos  -[<?  +  2  •* .   *  ~^ } 

^ 
and  if  this  be  put  equal  to  A  .  cos  (0  -f-  B)  we  get 

/        -2  i'  \ 

(I  -  «)  (1  -a)  -  2  v  a  a  (I  -  ,.)  (1  —  o).cosf2r  — J 

and  in  the  case  of  a  and  a  both  very  small 

f         £/  \  j 

A2  =  (  */  a  —  "J  «)a  +  4  .  V  a  a  .  sin  (  2  TT  —  I 

and  at  a  perpendicular  incidence,  where  t  =  t  ,  and  where  a  and  a  may  be  supposed  equal 

/       t  V 

A2  =  4  a  .  sin  I  2  T  —  I 

673.  Thus  we  see,  that  in  this  case  the  total  intensity  of  the  compound  reflected  wave  -f-  that  of  the  transmitted 
Not  con-  (Art.  662)  make  up  1,  the  intensity  of  the  incident  wave  ;  and  thus,  this  supposition  of  the  loss  or  gain  of  half 
trary  to  an  undulation  is  in  no  contradiction  with  the  law  of  the  conservation  of  the  vis  viva. 

In  fact,  however,  if  we  consider  the  mode  in  which  the  undulations  are  propagated,  at  the  limit  between  two 

>fl  6~4       media,  we  shall  see  nothing  contrary  to  dynamical  principles  in  the  loss  of  half  or  any  part  of  an  undulation  in 

Nor  to  the    the  transfer — for  it  cannot  be  supposed,  that  the  density  or  elasticity  of  the  ether  changes  abruptly  at  the  sur- 

undulatory    faces  of  media,  but  that  there  intervenes  some  very  minute  stratum  in  which  it  is  variable.     In  this  stratum, 

doctrine.      therefore,  the  length  of  an  undulation  is  neither  exactly  that  corresponding  to  the  denser,  nor  to   the   rarer 

medium,  but  intermediate,  and  of  a  magnitude  perpetually  varying.     Therefore  the  number  of  undulations  to  be 

reckoned  as  added  to  the  phase  of  the  ray  in  traversing  this  stratum,  will  differ  from  what  it  would  be  if  one 

medium  terminated,  and   the  other  commenced  abruptly.      Without  knowing  the  law  of  density,  the  limits 

between  which  it  undergoes  its  change,  or  the  exact  mode  in  which  the  partial  reflexion  of  a  wave  traversing  it 

.is  performed,  it  is  impossible  to  subject  the  point  to  strict  calculation,  we  must  rather  submit  to  be  taught  by 

experiment,  and  content  ourselves  with   such   conclusions   as  we  can  deduce  from   observation.     In  the  case 

before  us,  all  that  observation  teaches  us  is,  that  there  is  half  an  undulation  more  of  difference  in  the  phases  of 

two  rays  that  have  been  reflected  in  the  manner  last  considered,  than   in  those  of  the  two  whose  interference 

forms  the  transmitted  rays.     From  some  curious  experiments  of  Dr.  Young,  too,  we  may  gather  that  it  is  not  in 

all  cases  strictly  half  an  undulation  of  difference  to  be  reckoned,  but  rather  a  variable  fraction  depending  on  the 

nature  of  the  contiguous  media. 

The  formulae  of  Art.  672  show  that  it  is  only  in  the  case  of  perpendicular  incidence  that  the  tints  are  pure, 
675.       and  that  in  all  others,  and  especially   at  great  obliquities,  where  a  and  a  differ  considerably,  there  will  be  a 


LIGHT.  473 

Light,      dilution  of  white  light,  and  this  is  also  agreeable  to  experience.     At  a  perpendicular  incidence,  however,  the     Part  III. 
•"V*'  minima  of  each  homogeneous  colour  ought  to  be  absolutely  evanescent ;  so  that  if  we  were  to  remove  the  reflec-  v— v— •—•' 
tion  of  the  upper  surface  of  an  object  glass  laid  down  on  a  plate,  (or  use  a  prism,  so  as  to  prevent  its  reaching  Erperimen- 
the  eye,)  the  intervals  between  the  rings  in  homogeneous  light  ought  to  appear  absolutely  black.     In  the  New-  f"!"vc™' 
tonian  doctrine  this  should  not  be  the  case,  because  the  light  reflected  from  the  upper  surface  of  the  lamina  of  two  tneories 
included  air  should  still  remain  even  in  the  minima  of  the  rings.     This  then  affords  a  positive  means  of  deciding 
between  the  two  theories.     M.  Fresnel  describes  an  experiment  made  for  this  purpose,  and  states  the  result  to 
be  unequivocally  in  favour  of  that  of  undulations.     (Diffraction  dela  Lumiere,  p.  11.) 

§  V.     Of  the  Colours  of  Thick  Plates. 

Under  certain  circumstances  rings  of  colours  are  formed  by  plates  of  transparent  media  of  considerable  thick-      676 
ness.     The  circumstances  under  which  they  appear,  in  one  principal  case,  are  thus  described  by  Newton,  who 
first  observed  them,  and  who  has  applied  his  doctrine  of  the  fits  of  easy  reflexion  and  transmission  to  explain 
them,  with  singular  ingenuity. 

"  Admitting  a  bright  sunbeam  through  a  small  hole  of  one-third  of  an  inch  in  diameter  into  a  dark  room,  it  Newton's 
was  received  perpendicularly  on  a  concavo-convex  glass  mirror  one  quarter  of  an  inch  thick,  having  each  surface  ^{J^""1?111 
ground  to  a  sphere  of  six  feet  in  radius,  and  the  back  silvered.     Then  holding  a  piece  of  white  paper  in  the  m;rror 
centre  of  its  concavity,  having  a  small  hole  in  the  middle  of  it  to  let  the  sunbeam  pass,  and  after  reflexion  at 
the   speculum  to  repass  through  it,  the  hole  was  observed  to  be  surrounded  with  four  or  five  coloured  concen- 
tric rings  or  irises,  just  as  the  rings  seen  between  object-glasses  surround  their  central  spot — but  larger  and 
more  diluted  in  their  colours".  ..."  If  the  paper  was  much  more  distant   from  the  mirror,  or  much  less  than 
six  feet,  the  rings  became  more  dilute  and  gradually  vanished.1'.  ..."  The  colours  of  these  rings  succeeded  each 
other  in  the  order  of  those  which  are  seen  between  two  object  glasses,  not  by  reflected  but  by  transmitted  light, 

viz.  white,  tawny  white,  black,  violet,  blue,   greenish   yellow,  yellow,  red,  purple,"  &c "  The  diameters  of 

these  rings  preserved  the  same  proportion  as  those  between  the  object-glasses,  the  squares  of  the  diameters  of 
the  alternate  bright  and  dark  rings,  reckoning  the  central  white  as  a  ring  of  the  diameter  0,  forming  an  arith- 
metical progression,  beginning  at  0.  And  in  the  case  described,  the  diameter  of  the  bright  ring  measured 
respectively  0,  1-J£,  2-|,  2-J-J-,  3^.".  ..."  Lastly,  in  the  rings  so  formed  by  reflectors  of  different  thicknesses,  their 
diameters  were  observed  to  be  reciprocally  as  the  square  roots  of  the  thicknesses.  If  the  back  of  the  mirror  was 
silvered,  the  rings  were  only  so  much  the  more  vivid." 

These  various  phenomena,  and  a  variety  of  similar  ones,  some  of  more,  some  of  less  complexity,  according  to  677 
the  variation  of  the  distance,  and  obliquity  of  the  mirror,  and  the  curvature  of  its  surfaces,  Newton  has 
explained  very  happily,  by  considering  the  fits  of  easy  reflexion  and  transmission  of  that  faint  portion  of 
the  light  which  is  irregularly  scattered  in  all  directions  at  the  first  surface  of  the  glass,  and  which  serves  to 
render  it  visible.  But  for  this  explanation  we  must  refer  to  his  Optics,  as  our  object  here  is  more  particularly 
and  distinctly  to  show  what  account  the  undulatory  doctrine  gives  of  this  phenomenon,  which  has  hitherto  been 
passed  over  rather  cursorily,  not  without  some  degree  of  obscurity. 

There  is  no  surface,  however  perfectly  polished,   so  free  from   small  scratches  and  inequalities   as  not  to       678. 
reflect  and  transmit,  besides  those  principal  rays  which  obey  the  regular  laws  of  reflexion  and  refraction,  as  Principle  of 
dependent  on  the  general  surface,  other,  very  much  feebler,  portions  scattered  in   all  directions,  by  which  the  explanation 
surface  is  rendered  visible  to  an  eye  anywhere   placed,  but  most  copiously  in  and  about  the  direction  of  the  jn,tlie  un" 
regularly  reflected  and  transmitted  rays.     It  is  the  interference  of  these  portions,  scattered  at  the  first  surface  by  SyStac,'nry 
the  ray  in  passing  and  repassing  through  it,  nearly  in  its  own  direction,  that  the  rings  in  question  are  attributed 
in  the  undulatory  doctrine. 

Let  F  A  D,  E  B  G  be  the  parallel  surfaces  of  any  medium  exposed  perpendicularly  to  a  homogeneous  ray      679. 
emanating  from  a  luminous  point  C,  and  incident  at  A.     The  chief  portion  will  pass  straight  through  A,  and  be  Its  applica- 
reflected  back  from  B  to  A  again.     But  at  A  a  scattering  takes  place,  and  the  transmitted  ray  AB  is  accom-  ''?"• 
panied  by  a  diverging  cone  of  faint  rays  A  a,  A  b,  A  c,  &c.,  all  which  set  out  from  A  in  the  same  phase  of  their     g' 13 
undulations  with  the  principal  one  from   which  they  originate,  so  that  A  may  be  regarded  as  their  common 
origin.     Take  Q,  the  focus  of  rays  reflected  at  the  second  surface  conjugate  to  A  (if  the  surfaces  be  plane, 
Q  and  A  are  equidistant  from  B)  and  the  cone  of  scattered  rays,  with  the  regularly  reflected  ray  in  its  axis,  will 
after  reflexion  diverge  as  from  Q.     Again,  when  they  pass  into  the  air  again,  if  we  take  q  the  focus  conjugate 
to  Q  of  rays  refracted  at  the  surface  F  D,  they  will  after  refraction  diverge  from  q,  and  by  the  nature  of  foci  on 
the   undulatory  hypothesis,  the  undulations  will  be  propagated  in  the  air  as  if  they  had  a  common  origin  q 
placed  in  air  ;  because,  after  refraction,  the  waves  have  the  form  of  spheres  diverging  from  q,  and  therefore 
every  portion  of  their  surfaces  are  equidistant  from  q  ;  had  they,  therefore,  really  emanated  from  q,  as  separate 
rays,  they  must  at  the  moment  of  such  emanation  have  been  all  in  one  phase.     Now,  when  the  reflected  beam 
reaches  A  a  portion  of  it  will  again  be  scattered  in  a  cone,  having  the  regularly  transmitted  ray  A  G  in  its  axis  ; 
and  the  rays  A  O,  A  N,  AM,  &c.  of  this  cone  will  all  have  A  for  their  origin,  and  will  be  in  the  same  phase  at  their 
departure  from  A  with  the  ray  A  G ;   but  this  is  in  the  phase  it  would  have  had  as  emanated  from  q;  hence,  if 
we  consider  any  point  M  out  of  the  directly  transmitted  ray  A  G,  it  will  be  reached  at  once  by  a  wave  belonging 
to  each  diverging  cone,  the  one  along  q  M  from  q  and  the  other  along  AM  from  A,  and  the  difference  of  routes 
is  equal  to  </  A  -f-  A  M  —  g  M.     Therefore,  when  M  is  very  nearly  coincident  with  G,  this  is  very  small  and  at 
G  vanishes,  or  the  waves  are  in  exact  accordance.     As  M  recedes  from  G  it  increases,  and  when  it  becomes 

VOL.  ?v.  3  Q 


474  LIGHT. 

Light,      half  an  undulation,  the  waves  are  in  complete  discordance  and  annihilate  each  other,  and  so  on  alternately.    There 
•v"^  fore,  as  this  is  true  of  all  rays  in  conical  surfaces  round  A  G  as  an  axis,  equally  inclined  with  A  M,  q  M,  if  we  place 
awhile  screen  at  G,  it  will  appear  marked  with  alternate  dark  and  bright  rings  round  a  bright  centre.  To  deter- 
mine tlieir  diameters  we  need  only  put  </  A  -j-  A  M  —  <jr  M  ==  n  .  — ,  or,  if  we  take  q  A=  a,  A  G  =  r,  G  M  =  y, 


If  we  resolve  this  equation  neglecting  y2,  we  find 

y  =  V  n  .  \/   —  .  r  (a  +  r) 

•  ft 


which,  on  substituting  0,  1,2,  3,  &c.  in  succession  for  n,  shows  that  the  successive  diameters  of  the  alternate  dark 
and  bright  rings  are  in  the  progression  of  the  square  roots  of  those  numbers. 

680.  If  the  thickness  of  the  plate  be  small  compared  to  the  distance  of  the  screen,  a  will  also  be  small,  and  the 
Law  of  the    value  of  y  becomes 

diameters  of 
the  rings. 

which  shows  that  for  rays  of  a  given  refrangibility  the  diameters  of  the  rings  are  as  the  distance  of  the  screen 
directly,  and  the  square  root  of  the  thickness  of  the  plate  inversely. 

681.  Lastly,  the  diameter  of  a  ring  of  the  same  order  in  different  homogeneous  lights,  are  as  the  square  roots  of  the 
Of  their       lengths  of  their  undulations.     Now,  this  is  the  very  same  law  that  governs  the  diameters  of  the  rings  formed 
colours.        between  object-glasses.     Consequently,  if  instead  of  homogeneous  we  consider  white  light,  we  ought  to  have  a 

succession  of  coloured  rings  whose  tints  agree  precisely  with  the  transmitted  series  in  that  experiment. 

682.  But  the  rays  so  formed,  by  rays  scattered  from  a  single  point  A,  would  be  too  feeble  to  be  visible.     If,  how- 
Concentra-   ever,  we  suppose  the  surfaces  to  be  concentric  spheres  having  G  in  their  common  centre,  as  in  fig.  137,  then 
tion  of  the    any  rays  G  A,  G  A'  falling  on  any  points  whatever  of  their  surfaces  will  depict,  on  screens  G  M,  G  M'  respect- 
ring!,  from    jve]y  perpendicular  to  them  as  G,  equal  systems  of  rings  having  G  in  their  common  centre  ;   and,  when  the  arc 
the  surface.  A.  A'  's  not  verv  great.  the  screens  may  be  regarded  as  coincident  (for  in  that  case  B  M  —  M  A  =z  B  M'—  MA') 
Fig.  137.      and  the  rings  from  every  point  of  the  surface,  exactly  superposed  on  each  other,  and  being  thus  increased  in 

intensity  in  proportion  to  the  area  of  the  exposed  surface,  become  visible. 

683.  Now  this  is  exactly  Newton's  case,  for  the  sun  being  a  luminary  of  a  considerable  diameter,  the  hole  in  the 
Newton's      centre  of  the  spheres   may  be  regarded  as  a  portion  of  the   sun  of  that  size,  actually  placed  there.     Of  this, 
experiment   every  indivisible  point  may  be  regarded  as  the  origin  of  a  system  of  waves,  and  as   depicting  on  the  screen  its 

r"'d  d  own  set  °^  rmSs-  These,  were  the  hole  infinitely  small,  would  be  infinitely  more  clear  and  pure  in  their  tints 
than  the  transmitted  rings  between  object-glasses,  because  they  are  not  (as  in  those  rings)  diluted  with  the 
great  quantity  of  white  light  which  escapes  interference.  But  owing  to  the  size  of  the  hole,  their  centres  are 
not  exactly  coincident,  and  therefore  their  tints  mix  and  dilute  each  other,  and  that  the  more  the  larger  the 
hole  is. 

684.  If  c  be  the  thickness  of  the  glass,  since  Q  is  the  conjugate  focus  of  A,  on  the  surface  B  whose  radius  we  will 

call  r  -f  c  putting  G  A  =  r,  we  have,  by  Art.  249,  B  Q  =  — .  c,  A  Q  =    — — ;  and,  by  Art.  24S 

T  —  c  r  —  c 

2  C  T 

A  o=  a  = — ; — — ,  taking  ufor  the  refracti/e  index;  and  when  c  is  small  compared  with  r,  we  get 

2  c  —  /» (r  -f-  c) 

2  C    •   «  =  r 

2  '   c 

showing  that  the  diameters  of  the  rings  are  in  this  case  in  the  subduplicate  ratio  of  the  refractive  index  of  the 
glass  directly,  and  of  its  thickness  inversely. 

89000 


,  —      /7      5T 

»-  Vf  •- 


685  ^  we  re<^uce  ^is  value  to  numbers,  taking  /»  =  —  ,  n  =:  4,  r  =  6  feet  =  7£'  inches,  and  X  =  --  =  the 


length  of  an  undulation  for  yellow  rays  -  nearly,  we  find,  for  the   diameter  of  the  second  bright  ring  in 

yoooo 

yellow  light,  (which  corresponds  to  the  brightest  part  of  the  same  ring  in  white,) 

-  4  =  2-35, 


which  agrees  almost  precisely  with  Newton's  measure  2$,  or  2'375. 

686  When  the  mirror  is  inclined  to  the  incident  beam  the  phenomena  become  more  complicated,  and  have  been 

'  Case  of        elegantly  described  by  Newton,  (Optics,  book  ii.  part  iv.  obs.  10.)     In  this  case,  the  axes  of  the  two  interfering 

oblique         cones  of  scattered  rays,  which   are  always  the  incident  and   reflected  rays,  are  no  longer  coincident.     But  the 

incidence.     same  principles  apply  equally  to  this  case  in  all  other  respects,  and  the  reader  may  exercise  himself  in  tracing 

PH  their  consequences. 

ot>Mrved  by  "^he  ^u'ce  de  Chaulnes  found  similar  rings  to  be  exhibited  when  the  surface  of  the  mirror  was  covered  with 
the  Duke  of  a  thin  film  of  milk  dried  on  it,  so  as  to  make  a  delicate  semitransparent  coating,  or  even  when  a  fine  gauze  or 
Chaulnes  muslin  was  stretched  before  it;  see  the  account  of  his  experiments  in  the  Mem.  Acad.  Sci.  Paris,  1705  ;  and 
and 


LIGHT.  475 

Sir  William  Herschel  (Phil.  Trans.  1807)  describes  a  pleasing  experiment,  in  which  rings  were  produced  by    Part  III. 
^  strewing  hair  powder  in  the  air  before  a  metallic  mirror  on  which  a  beam  of  light  is  incident,  and  intercepting  V<""~V^' 
the  reflected  ray  by  a  screen.     The  explanation  of  these  phenomena  seems,  however,  to  depend  on  other  appli-  ^ir  w- 
cations  of  the  general  principle,  and  will  be  better  conceived  when  we  come  to  speak  of  the  colours  produced 
by  diffraction. 

Dr.  Brewster,  in  the  Transactions  of  the  Royal  Society  of  Edinburgh,  has  described  a  series  of  coloured  fringes      " 
produced  by  thick  plates  of  parallel  glass,  which   afford  an  excellent  illustration  of  the  laws  of  periodicity  ^f " 
observed  by  the  rays  of  light  in  their  progress,  whether,  as  in  the  Newtonian  doctrine,  we  consider  them  as  sub-  fringes  seen 
jected  to  alternate  fits  of  easy  reflexion  and  transmission,  or,  as  in  the  undulatory  hypothesis,  as  passing  through  in  thick 
a  series  of  phases  of  alternately  direct  and  retrograde  motions  in  the  particles  of  ether,  in  whose  vibrations  they  plates, 
consist.     We   may  here  remark,  once  for  all,  that  the  explanations  which  the   undulatory  doctrine    affords   of 
phenomena  of  this  description,  may,  for  the  most  part,  be  translated  into  the  language  of  the  rival  hypothesis  ; 
so  as  to  afford,  with  more  or  less  plausibility  and  occasional  modifications,  a  result  corresponding  with  observa-         , 
tion.     It  is  not,  therefore,  among  phenomena  of  this  class  that  we  must  look  for  the  means  of  deciding  between 
them.     We  shall  adopt,  therefore,  in  the  remainder  of  this  essay,  the  undulatory  system,  not  as  being  at  all 
satisfied  of  its  reality  as  a  physical  fact,  but  regarding  it  as  by  far  the  simplest  means  yet  devised  of  grouping 
together,  and  representing  not  only  all  the  phenomena  explicable  by  Newton's  doctrine,  but  a  vast  variety  of 
other  classes  of  facts  to  which  that  doctrine  can  hardly  be  applied  without  great  violence,  and  much  additional 
hypothesis  of  a  very  gratuitous  kind. 

The  fringes  in  question  are  seen  when  two  parallel  plates  of  glass  of  exactly  equal  thickness  (portions  of  the  ggg 
same  plate)  are  slightly  inclined  to  each  other,  (at  any  distance,)  and  through  them  both,  at  nearly  a  perpen-  Described 
dicular  incidence,  a  circular  luminary  of  1°  or  2°  in  diameter  (a  portion  of  the  sky,  for  instance)  is  viewed. 
There  will  in  this  case  be  seen,  besides  the  direct  image,  a  series  of  lateral  images  reflected  between  the  glasses, 
and  growing  fainter  and  fainter  in  succession  as  they  are  formed  by  2,  4,  6,  or  more  internal  reflexions; 
and  of  which  all  hut  the  first  is  so  faint  as  scarcely  to  be  visible,  except  in  very  strong  lights.  The  direct  image 
is  colourless ;  but  the  reflected  one  is  observed  to  be  crossed  with  fifteen  or  sixteen  beautiful  bands  of  colour, 
parallel  to  the  common  section  of  the  surfaces  of  the  plates.  Their  breadth  diminishes  rapidly  as  the  inclination 
of  the  plates  increases.  When  the  plates  employed  were  0.121  inch  in  thickness,  and  inclined  at  an  angle  of 
1°  11'  to  each  other,  the  breadth  of  each  fringe  measured  26'  50",  and  at  all  other  inclinations  their  breadth  was 
inversely  as  the  inclination.  At  oblique  incidences  its  fringes  are  seen  when  the  plane  of  incidence  is  at  right 
angles  to  the  principal  section  of  the  plates,  but  are  at  their  maximum  of  distinctness  when  parallel  to  it. 

To  understand  their  production,  let  us  call  the  surfaces  of  the  plates  in  order,  reckoning  from  that  on  which      690 
the  incident  light  first  falls,  A,  a,  B,  6;  and  let  us  consider  a  ray,  or  system  of  waves  emanating  from  a  common  Explained, 
origin  at  an  infinite  distance.     Then,  when  this  ray  falls  on  the  plates  it  will  at  every  surface  undergo  a  partial 
reflexion,  and  the  remainder  will  be  transmitted ;  each  of  the  several  portions  will  be  again  subdivided  when- 
ever it  meets  either  surface.     So  that  either  image  will,  in  fact,  consist  of  several   emergent  rays,    parallel  in 
their  final  directions,  but  which  have  traversed  the  glasses  by  very  different  routes.     Thus  the  direct  or  principal 
image  will  consist  of 

1.  The  chief  portion  of  the  whole  incident  light,  refracted  at  A,  at  a,  at  B,  and  at  b,  and  emergent  parallel 
to  the  incident  ray,  which  we  will  represent  by  A  a  B  b. 

2.  A  portion  refracted  at  A,  reflected  at  a,  reflected  again  at  A,  refracted  again  at  a,  at  B  and  at  b,  and 
emergent  parallel  to  the  incident  beam.     This  we  will   denote  thus,  A  a'  A.'  a  B  b ;   the  letters  denoting  the 
surfaces,  the  accent  reflexion,  and  its  absence  refraction. 

3.  A  portion  which  has  undergone  two  similar  reflexions  in  the  interior  of  the  second  plate,  and  which  in 
the  same  manner  may  be  represented  by  A  a  B  b'  B'  6. 

4.  Other  portions  which  have  undergone  respectively  four,  six,  &c.  reflexions  to  infinity  within  either  of  the 
plates,  and  which  may  be  represented   by  such  combinations  as  A  a'  A' a'  A'aB  b,  A  a  B  b'E'b'Wb,  or,  for 
brevity,  by  A  (a1  A')2  a  B  b,  A  a  B  (b'  B')2  b,  &c. ;    but  these  latter  portions  are  too  faint  to  have  any  sensible 
influence  on  the  light  of  the  direct  image  with  which  they  are  confounded. 

The  first  lateral  reflected  image  will  consist  of  four  principal  portions  which  have  undergone  two  reflexions      591 
each,  viz. 

AaB'o'Bi;        Aa  B'a  A'o  B  b;        AaB&'B  a'B  b  ;        AaB  b' a  A'aB  b; 

all  which  will  emerge  parallel.  Besides  these  there  are  infinite  others,  formed  by  a  greater  number  of  reflexions, 
and  by  the  portions  A  a' A' a  of  the  incident  beam  reflected  within  the  first  glass;  but  these  are  all  too  faint 
materially  to  affect  the  image  in  question,  which  therefore  we  may  regard  as  composed  solely  of  the  four  rays 
just  enumerated.  Now  if  we  cast  our  eye  on  the  figure,  (138,)  we  see  the  course  pursued  by  each  of  these  pig  133 
portions  1,  2,  3,  4  ;  and  it  is  evident  that  the  first  portion  has  traversed  the  thickness  twice,  and  the  interval 
between  the  glasses  three  times,  or  nearly;  neglecting  at  present  all  consideration  of  the  inclination  of  the 
plates  2  t  -(-  3  i.  In  like  manner,  the  portion  2  will  have  traversed  4 1  -f-  3  i ' ;  the  portion  3,  4  t  -j-  3  i;  and  the 
portion  4,  6  t  -}-  3  i.  Hence  it  appears  that  the  portions  1  and  4  differ  in  their  routes  by  nearly  four  times  the 
thickness  of  the  glass,  and  can  therefore  produce  no  colours ;  but  the  other  portions,  at  a  perpendicular  inci- 
dence, would  not  differ  at  all,  and  at  very  small  inclinations  of  the  plates,  and  of  the  incident  ray,  will  only  differ 
by  reason  of  the  small  differences  of  the  inclinations  at  which  they  traverse  their  respective  thicknesses  and 
intervals.  They  will,  therefore,  interfere  so  as  to  produce  colour ;  and  this  will  be  dependent  on  the  interval 
of  retardation  of  one  ray  behind  the  other,  arising  from  the  varying  obliquity  of  the  ray  which  enters  the  eye. 

Now  when  we  look  at  a  luminous  image  of  sensible  magnitude,  the  rays  by  which  we  see  its  several  points      "92. 

3(j2 


476 


LIGHT. 


are  incident  in  all  planes,  and  at  all  inclinations.     Hence,  the  image  seen  will  appear  of  different  colours  in  its   P»rt  Ill- 
different  points,  and  the  disposition  of  these  colours  will  follow  the  law,  whatever  it  be,  which  regulates  the  v"- "v^** 
Isochro-       interval  of  retardation.     The  colours,  therefore,  will  be  arranged  in  bauds,  circles,  or  other  forms,  according  to 
j"*""  i'nes   tne  f°rm  °f  *ne  curves  arising  geometrically  from  the  consideration  of  equal  intervals  of  retardation  prevailing 
in  every  point  of  their  course.     Such  curves,  now  and  hereafter,  we    shall  term   isochromatic  lines,  or  lines  of 
equal  tint,  measuring  in  all   cases  the  lint  numerically  by  the  number  of  undulations,  or  parts  of  an  undulation 
of  mean  yellow  light  to  which  the  interval  of  retardation  is  equal. 

Let  us,  then,  first  consider  the  case  when  the  ray  is  incident  in  a  plane  perpendicular  to  the  common  section. 
In  this  case,  fig.  139,  let  K  L  M  N  be  a  ray  formed  by  the  union  of  two  rays  SAoB&IKLandSCEFGHKL, 
whose  courses  through  the  system  are  similar  to  2  and  3,  fig.  138.  Draw  AD  perpendicular  to  S  C,  then  will 
the  interval  of  retardation  be  equal  to 


Light 


defined. 


693. 
Fig.  139. 


=  DC+  (EF-aB)  +  (FG  -  IK)  +  2  (K  H  -  B  i), 

the  first  three  terms  being  performed  in  air,  the  last  in  glass.  Now,  without  entering  into  a  trigonometrical 
calculation,  it  is  evident  that  this  will  be  very  small  at  a  perpendicular  incidence,  and  will  increase  rapidly  as 
the  angle  of  incidence  varies ;  and  that  (the  inclination  of  the  plates  remaining  constant)  it  will  increase  by 
nearly  equal  increments,  as  the  incidence  varies  by  equal  changes  from  0  on  either  side  of  the  perpendicular. 
Therefore,  in  a  direction  at  right  angles  to  the  common  section  of  the  surfaces  the  tints  will  vary  rapidly, 
increasing  on  either  side  of  the  perpendicular  incidence  ;  and  at  very  moderate  obliquities  on  either  side, 
the  interval  of  retardation  will  become  too  great  for  the  production  of  colour.  On  the  other  hand,  if  we 
conceive  the  rays  S  A,  S  C,  to  be  incident  in  a  plane  very  nearly  parallel  to  the  principal  section,  then  will  the 
points  K  and  G  be  situated,  not.  as  in  the  figure,  at  different  distances  from  P,  but  at  very  nearly  the  same ;  so 
that  (whatever  be  the  incidence)  K  I  will  very  nearly  equal  G  F,  and  for  the  same  reason  F  E  will  very  nearly 
equal  a  B.  •  Moreover,  in  this  case  G  K  will  be  very  nearly  equal  to  F  I,  and  the  angles  of  internal  incidence 
will  be  also  very  nearly  equal,  so  that  H  G  -f-  G  K  will  differ  very  little  from  B  b  +  b  I,  and  I  B  will  be  very 
nearly  equal  to  G  K,  and  therefore  to  I  F,  so  that  the  point  F  will  almost  exactly  coincide  with  B,  and  the  rays 
S  A  a  B,  SCEF  will  coincide  almost  precisely,  making  D  C  =  0  ;  and  these  approximate  equalities  and  coin- 
cidences will  continue  for  great  variations  in  the  angle  of  incidence,  provided  the  plane  of  incidence  be  unaltered. 
The  interval  of  retardation,  then,  will  in  this  case  depend  very  little  on  the  angle  of  incidence ;  so  that  in  a 
direction  parallel  to  the  common  section  of  the  surfaces,  the  tints  will  vary  but  little.  Hence  it  appears  that 
they  will  be  arranged  in  the  manner  described  by  Dr.  Brewster,  viz.  in  fringes  parallel  to  that  line.  Their 
general  analytical  expression  is,  however,  rather  too  complex  to  be  here  set  down,  though  very  easily  investigated 
from  what  has  been  said. 

By  intercepting  the  principal  transmitted  beam  in  the  direct  image,  and  receiving  on  the  eye  only  those 
portions  of  the  rays  going  to  form  it  whose  curves  are  as  in  fig.  140,  or  the  portions  A  a'  A' a  B  b,  and 
AoBft'B'6,  Dr.  Brewster  succeeded  in  rendering  visible  a  set  of  coloured  fringes,  which  in  general  are  diluted 
and  concealed  in  the  overpowering  light  of  the  direct  beam.  They  originate  evidently  in  the  interference  of 
these  two  rays,  whose  courses  are  each  represented  by  4  t  4  i,  and  would  therefore  be  strictly  equal  were  the 
plates  exactly  parallel.  Their  theory,  after  what  has  been  said,  will  be  obvious  on  inspection  of  the  figure,  as 
well  as  those  of  all  the  rest  of  the  systems  of  fringes  which  he  has  described  in  that  highly  curious  and  inte 
resting  memoir. 

Mr.  Talbot  has  observed,  when  viewing  films  of  blown  glass  in  homogeneous  yellow  light,  and  even  in 
common  daylight,  that  when  two  films  are  superposed  on  each  other,  bright  and  dark  stripes,  or  coloured  bands 
glass  films  anc^  fringes  °f  irregular  forms,  are  produced  between  them,  though  presented  by  neither  separately.  These  are 
obviously  referable  to  the  same  principle,  the  interference  taking  place  here  between  rays  respectively  twice 
reflected  within  the  upper  lamina,  and  once  reflected  at  the  upper  surface  of  the  lower  lamina,  or  else  between 
rays  one  of  which  is  thrice  reflected  in  the  mode  represented  by  A  a  B'  a'  B'  a  A,  and  the  other  in  that  repre- 
sented by  A  a  B' a  A' a' A,  the  interval  between  the  glasses  being  supposed  to  be  exactly  equal  to  the  thickness 
of  the  upper  one  in  both  cases,  a  condition  which  is  sure  to  obtain  somewhere  when  the  laminae  are  curved.  A 
still  more  curious  and  delicate  case  of  the  production  of  similar  fringes  has  been  noticed  by  Professor  Amici,  to 
take  place  when  two  of  the  blue  feathers  of  the  wing  of  the  Papilio  Idas  (a  species  of  butterfly)  are  laid  on 
each  other  in  the  field  of  his  powerful  and  exquisite  microscopes.  These  feathers  he  describes  as  small  plates 
of  perfect  transparency,  and  uniformly  and  delicately  striated  over  their  whole  surface.  The  fringes  in  question 
are  formed  between  them,  and  vary  in  breadth,  form,  and  situation,  according  to  the  manner  in  which  the 
feathers  are  superposed.  Their  origin  seems  to  be  independent  of  the  stria:  however,  and  is  easily  understood 
on  the  principles  above  explained.  The  same  may  be  said  of  the  colours  observed  by  Mr.  Nicholson  in  combi- 
nations of  parallel  glasses  of  unequal  thickness.  Suppose,  for  instance,  that  instead  of  the  plates  having 
exactly  equal  thicknesses,  their  thicknesses  I,  tf  differ  by  a  very  minute  quantity,  then  the  course  of  the  rays 
A  a'  A'  a  B  6  and  A  a  B  6'  B'  6  will  (at  a  perpendicular  incidence)  be  respectively  3  t  -f-  i  -j-  f  and  t  +  i  -f  3  V. 
(supposing  the  plates  strictly  parallel,)  and  the  difference  of  their  routes  is  2  t  —  2  if  ;  so  that  if  this  be  exceed- 
ingly minute,  colours  will  arise,  or,  if  not,  may  be  produced  by  a  slight  inclination  of  the  plates  to  each  other, 
and  so  of  an  infinite  variety  of  cases  which  may  arise. 


694. 
Fig.  140. 


695 

Fringes 
between 


LIGHT.  477 

ught.  i 

§  VI.     Of  the  Colours  of  Mixed  Plata. 

The  colours  hitherto  described  have,  been  referred  to  the  interference  of  rays  rigorously  coincident  with  each      gg^ 
other  throughout  their  whole  course,  after  the  point  where  they  begin  to  be  superimposed.     Such   interfering  interference 
rays,  or  systems  of  waves,  being  united  into  a  point  on  the  retina,  that  point  is  agitated  by  the  sum  or  difference  of  rays  not 
of  their  actions,  and  the  sensation  produced  is  according.     But  if  this  coincidence  be  only  approximate,  as,  if stl 
two  systems  of  waves  be  propagated  from  origins  so  nearly  coincident  in  angular  situation  from  the  eye,  that  co 
their  images  formed  on  the  retina  shall  be  too  close  to  be  distinguished  by  the  mind  from  the  image  of  a  single 
point,  the  impressions  produced  will  still  be  confounded  together;    or  rather,  we  ought  to  say,  the  mechanical 
action  on  one  point  will  be  propagated  through  the  substance  of  the  retina  to  the  other,  and  a  sensation  cor 
responding  to  their  mean  or  average  effect  will  be  produced.     If,  then,  the  rays   concentered   on   contiguous 
points  of  the  retina  be  in  exact  discordance,  and  of  equal  intensity,  a  mutual  destruction  will  take  place,  as  if 
they  fell  on  one  mathematical  point ;  if  in  exact  accordance,  they  will  increase  each  others  effects,  and  so  for  the 
intermediate  states. 

To  apprehend  this  more  fully,  we  must  consider  that  the  impression  of  light  appears  to  spread  on  the  retina      697. 
to  a  certain  extremely  minute  distance  all  around  the  mathematical  focus  of  the  rays  concentered  by  the  lenses  Irradiation, 
of  the  eye.     Thus  the  image  of  a  star  is  never  seen  as  a  point,  but  as  a  disc  of  sensible  size,  and  that  the  larger 
as  the  light  is  stronger.     Thus,  too,  the  bright  part  of  the  new  moon  is  seen,  as  it  were,  larger  than  the  faintly 
illuminated  portion  of  its  disc  projecting  beyond  it  as  an  acorn  cup  beyond  the  fruit,  &c.     This  effect  is  termed 
irradiation,  and  is  manifestly  the  consequence  of  an  organic  action  such  as  we  have  described. 

It  follows  from  this,  that  when  waves  emanate  from  origins  undistinguishably  near,  they  may  be  regarded  in  698. 
their  effects  on  the  eye  as  emanating  from  origins  strictly  in  one  and  the  same  right  lines,  the  direction  of  the 
joint  ray ;  and  the  laws  of  their  interferences  will  be  precisely  the  same,  considered  in  their  effect  on  vision,  as  if 
the  lenses  of  the  eye  were  away,  and  the  retina  were  a  mere  screen  of  white  paper,  on  a  single  physical  point  of 
which  (viz,  the  point  where  the  images  concentered  by  the  lenses  would  have  fallen)  the  interfering  undulations 
propagated  simultaneously  from  the  two  origins  fell,  and  agitated  it  with  a  vibration  equal  to  their  resultant. 

This  premised,  we  are  in  a  condition  to  appreciate  the  explanation  afforded  by  the  undulatory  doctrine  of  the  699. 
phenomena  of  mixed  plates.  They  were  first  noticed  (says  Dr.  Young)  by  him  "  in  looking  at  a  candle  through  two  Phenomena 
pieces  of  plate  glass  with  a  little  moisture  between  them.  He  thus  observed  an  appearance  of  fringes  resembling  °*  mixe(' 
the  common  colours  of  thin  plates ;  and  upon  looking  for  the  fringes  by  reflexion,  found  that  the  new  fringes  p ' 
were  always  in  the  same  direction  as  the  others,  but  many  times  larger.  By  examining  the  glasses  with  a 
magnifier,  he  perceived,  that  wherever  the  fringes  were  visible,  the  moisture  was  intermixed  with  portions  of  air 
producing  an  appearance  similar  to  dew."  "  It  was  easy  to  find  two  portions  of  light  sufficient  for  the  produc- 
tion of  these  fringes;  for  the  light  transmitted  through  the  water  moving  in  it  with  a  velocity  different  from 
that  of  light  passing  through  the  interstices  filled  only  with  air,  the  two  portions  would  interfere  with  each  other 
and  produce  effects  of  colour  according  to  the  general  law.  The  ratio  of  the  velocities  in  water  and  air  is  that 
of  three  to  four ;  the  fringes  ought  therefore  to  appear  where  the  thickness  is  six  times  as  great  as  that  which 
corresponds  to  the  same  colour  in  the  common  case  of  thin  plates ;  and  upon  making  the  experiment  with  a 
plane  glass  and  a  lens  slightly  convex,  he  found  the  sixth  dark  circle  actually  of  the  same  diameter  as  the  first 
in  the  new  fringes.  The  colours  are  also  easily  produced  when  butter  or  tallow  is  substituted  for  water,  and 
the  rings  then  become  smaller  in  consequence  of  the  greater  refractive  density  of  the  oils ;  but  when  water  is 
added  so  as  to  fill  up  the  interstices  of  the  oil,  the  rings  are  very  much  enlarged  ;  for  here  the  difference  of 
velocities  in  water  and  in  oil  is  to  be  considered,  and  this  is  much  smaller  than  the  difference  between  air  and 
water.  All  these  circumstances  are  sufficient  to  satisfy  us  of  the  truth  of  the  explanation,  and  is  still  more 
confirmed  by  the  effect  of  inclining  the  plates  to  the  direction  of  the  light;  for  then,  instead  of  dilating  like  the 
colours  of  thin  plates,  these  rings  contract,  and  this  is  the  obvious  consequence  of  an  increase  of  the  lengths 
of  the  paths  of  the  light  which  now  traverses  both  media  obliquely,  and  the  effect  is  everywhere  the  same  as 
that  of  a  thicker  plate.  It  must,  however,  be  observed,  that  the  colours  are  not  produced  in  the  whole  light 
that  is  transmitted  through  the  media ;  a  small  portion  only  of  each  pencil  passing  through  the  water  contiguous 
to  the  edges  of  the  particle  is  sufficiently  coincident  with  the  light  transmitted  through  the  neighbouring  portions 
of  air  to  produce  the  necessary  interference  ;  and  it  is  easy  to  show  that  a  considerable  portion  of  the  light  that 
is  beginning  to  pass  through  the  water  will  be  dissipated  laterally  by  reflexion  at  its  entrance,  on  account  of 
the  natural  concavity  of  the  surface  of  each  portion  of  the  fluid  adhering  to  the  two  surfaces  of  the  glass,  and 
that  much  of  the  light  passing  through  the  air  will  be  scattered  by  refraction  at  the  second  surface.  For  these 
reasons  the  fringes  are  seen  when  the  plates  are  not  directly  interposed  between  the  eye  and  the  luminous 
object."  (Young,  Phil.  Trans.  1802  ;  Account  of  some  Cases  of  the  Production  of  Colours.)  To  see  the 
phenomena  to  advantage,  we  may  add,  it  is  only  necessary  to  rub  up  a  little  froth  of  soap  and  water  almost  dry 
between  two  plane  glasses,  and  hold  them  at  a  distance  from  the  eye  between  it  and  a  candle,  or  the  reflexion 
of  the  sun  on  any  polished  convex  object.  If  two  slightly  convex  glasses,  or  a  plane  and  a  convex  one  be  used, 
the  colours  are  seen  arranged  in  rings. 


478  LIGHT. 

Light.  Part  III. 

§  VII.  Of  the  Colours  of  Fine  Fibres  and  Striated  Surfaces. 

If  two  points  supposed  capable  of  reflecting  light  in  all  directions  (as  two  infinitely  small  spheres,  &c.)  be  so 

snce  near  eacfj  ot)jer  as  to  appear  to  the  eye  as  one,  and  if  rays  from  a  common  origin  reflected  from  them  reach  the 

rlecte'd  from  eye>  *^ey  w'"  interfere  ;   and  if  the  light  be  homogeneous,  its  intensity  will  vary  periodically,  with  an  interval  of 

points  or      retardation  corresponding  to  the  difference  of  their  paths  ;    if  white,  the  colour  of  the  mixed  reflected  ray  will  be 

lines  very     the  same  as  if  it  had  been  transmitted  through  a  plate  of  air  of  a  thickness  equal  te  that  difference,  but  deprived 

«ear  each     of  jjs  djlutjng.  white.     Suppose  two  exceedingly  fine  cylindrical  polished  fibres  to  be  placed  at  right  angles  to 

Fi/'wi       ^e  'me  °^  s'8'ht>  and  parallel  to  each  other,  as  in  fig.  141,  as  ABC,  a  b  c  ;  and  let  S  be  a  luminous  point  very 

distant  with  respect  to  the  interval  of  the  fibres,  and  E  the  eye,  placed  so  as  to  receive  the  reflected  rays  B  E, 

6  E,  which,  by  supposition,  are  near  enough  to  interfere.     Then  the  differences  of  phases  of  the  rays  on   the 


(S  6  +  6  E;>  -  (S  B  +  B  E)  bx  +  by 

retina  is  evidently  equal  to  2  ir  x  -  -  •  -  —  =  2  v  .  -  -  -  —  ,  supposing  B  x  and  B  y 

\  n, 

perpendicular  to  S  6  and  ft  E.  If,  then,  we  suppose  I  and  i  to  be  the  angles  of  incidence  of  the  rays  S  B,  E  B 
on  the  plane  in  which  the  axes  of  the  two  cylinders  AC,  ac  lie,  and  put  B  6  their  distance  equal  to  a,  we  have 
for  the  difference  of  phases 

a 

2  TT  .    —  .  (sin  I  +  sin  i). 
X 

Hence,  if  a  remain  the  same,  this  will  vary  with  the  obliquity  both  of  the  incident  and  reflected  ray  to  the  plane 
of  the  axes  of  the  fibres  ;  and,  therefore,  if  that  plane  be  turned  about  an  axis  parallel  to  the  fibres,  a  succession 
of  colours  analogous  to  the  transmitted  series  of  those  of  their  plates,  but  much  more  vivid,  will  be  seen,  as  if 
reflected  on  them. 

701.  Any  extremely  fine  scratch  on  a  well  polished  surface  may  be  regarded  as  having-  ..  concave,  cylindrical,  or, 
Colours  of    at  least,  a  curved  surface  capable  of  reflecting  the  light  equally  in  all  directions;  this  is  evident,  for  it  is  visible 
scratches  on  jn  a]j  djrectjOns.     Two  such  scratches,  then,  drawn  parallel  to  each  other,  and  then  turned  round  an  axis  parallel 

to  both  in  the  sunshine,  ought  to  affect  the  eye  in  succession  with  a  series  of  colours  analogous  to  those  of  thin 
plates.  This  is  really  the  case.  Dr.  Young  found,  on  examining  the  lines  drawn  on  glass  in  Mr.  Coventry's 
micrometric  scales,  each  of  them  to  consist  of  two  or  more  finer  lines  exactly  parallel,  and  at  a  distance  of  about 
one  10,000th  of  an  inch.  Placing  the  scale  so  as  to  reflect  the  sun's  light  at  a  constant  angle,  and  varying  the 
inclination  of  the  eye,  he  found  the  brightest  red  to  be  produced  at  angles  whose  sines  were  in  the  arithmetical 
progression  1,  2,  3,  4. 

702.  In  the  beautiful  specimens  of  graduation  on   glass  and  steel   produced  by  Dr.  Wollaston,   Mr.  Barton,  and 
3*  systems    $[.  Fraunhofer,  single  lines  exactly  parallel  to  each  other,  and  distant  in  some  cases  not  more  than  one  10,000th 

of  an  inch,  and  at  precisely  equal  intervals,  are  drawn  with  a  diamond  point.  If  the  eye  be  applied  close  to  a 
parallel  reflecting  or  refracting  surface  so  striated,  so  as  to  view  a  distant,  small,  bright  light  reflected  in  it,  it  will  be  seen 
lines.  accompanied  with  splendid  lateral  spectra,  which  evidently  originate  in  this  manner.  They  are  arranged  in  a 

straight  line  passing  through  the  reflected,  colourless  image,  and  at  right  angles  to  the  direction  of  the  striae. 
Their  angular  distances  from  each  other,  the  succession  of  their  colours,  and  all  their  other  phenomena,  are  in 
perfect  agreement  with  the  above  explanation.  Their  vividness  depends  on  the  exact  equality  of  distance 
between  the  parallel  lines,  which  causes  the  lateral  images  produced  by  each  pair  to  coincide  precisely  in 
distance  from  the  principal  image,  and  thus  to  produce  a  multiplied  effect.  If  the  distance  of  the  lines  be 
unequal,  the  images  from  different  pairs,  not  coinciding,  blend  their  colours,  and  produce  a  streak,  or  ray  of 
white  light.  This  is  the  origin  of  the  rays  seen  darting,  as  it  were,  from  luminous  objects  reflected  on  irregularly 
polished  surfaces.  These  colours  may  be  transferred,  by  impression  from  the  surface  originally  graduated,  to 
sealing  wax,  or  other  soft  body  ;  or  from  steel,  by  violent  pressure,  to  softer  metals.  It  is  in  this  way  that  those 
beautiful  striated  buttons  and  other  ornaments  are  produced,  which  imitate  the  splendour  and  play  of  colours 
of  the  diamond. 

703.  Dr.  Young  has  assimilated  the  colour  thus  produced  when  a  beam  of  white  light  strikes  on  a  succession  of 
Alleged        parallel  equidistant  lines,  to  the  musical  tone  heard  when  any  sudden  sound  is  echoed  in  succession  by  a  series 

^pn       of  equidistant  bars  having  flat  surfaces  situated  in  a  direction  perpendicular  to  the  line  in  which  they  are  arranged, 

colours"^     f°r  instance,  an  iron  railing.     It  is  evident  that  such  echoes  will  reach  the  ear  in  succession,  at  precisely  equal 

striated        intervals  of  time,  each  being  equal  to  the  time  taken  by  sound  to  traverse  twice  the  space  separating  the  bars  ; 

surfaces        and  thus  producing  on  the  ear,  if  the  bars  be  sufficiently  numerous,  the  effect  of  a  musical  sound.  (Phil.  Trans. 

ande.cerum    1801  ;    On  the  Theory  of  Light  and  Colours.)     This   explanation,  however,   appears    to    us,  we  confess,  more 

es^on-     ingenious  than  satisfactory.     The  pitch  of  the  musical  tone  produced  by  the  echoes  is  independent  of  the  sound 

sidered.        echoed,  which  may  be  a  single  blow,  or  a  noise,  (i.  e.  a  sound  consisting  of  non-periodic  vibrations,)  and  requires 

for  its  production  a  number  of  echoing  bars  sufficient  to  prolong  the  echoes  a  sensible  time.     On  the  other  hand, 

the  light  reflected  from  parallel  stris  depends  for  its  colour  wholly  on  the  incident  ray,   being  red  in  red  light, 

yellow  in  yellow,  &c.  ;  and  is  produced  equally  well  from  two  or  from  twenty,  as  from  a  million  of  such  reflecting 

lines.     The  intensity,  not  the  colour,  —  the  magnitude,  not  the  frequency  of  the  impression  made  on  the  retina  by 

the  reflected  rays,  is  modified  by  their  interference.     We  think  it  necessary  to  point  out  this  defect  in  the  illus- 

tration in  question,  inasmuch  as  it  has  become  popular  for  its  ingenuity,  and  primd  facie  plausibility  ;  while,  in 

reality,  it  is  calculated  to  give  very  erroneous  impressions  of  the  analogy  between  sound  and  light. 


LIGHT.  479 

A  single  scratch  or  furrow  in  a  surface  may,  ns  that  eminent  philosopher  has  himself  remarked,  produce  colours    Part  III. 

•>  by  the  interference  of  the  rays  reflected  from  its  opposite  edges.     A  spider's  thread  is  often  seen  to  gleam   in  ^^—^^-^ 

the  sunshine  with  the  most  vivid  colours.     These  may  arise  either  from  a  similar  cause,  or  from  the  thread  itself      704. 

as  spun  by  the  animal,  consisting  of  several,  agglutinated  together,  and  thus  presenting  not  a  cylindrical,  but  a  Colours  of 

furrowed  surface.  weftc'* 

The  phenomena  exhibited  by  light  reflected  from  and  refracted  through  the  polished  surface  of  mother  of  ^.Q^  ' 
pearl,  are,  no  doubt,  referable  in  great  measure  to  the  same  principle,  so  far  as  they  depend  on  the  structure  Of  moth'er 
of  the  surface.  Dr.  Brewster  has  described  them  in  a  most  curious  and  interesting  Paper,  (published  in  the  Of  pearl 
Phil.  Trans.  1814,  p.  397  ;)  and  a  writer  in  the  Edinburgh  Philosophical  Journal,  vol.  ii.  p.  117,  has  added 
some  further  particulars  illustrative  of  the  curious  and  artificial  structure  of  this  singular  body.  Every  one 
knows  that  mother  of  pearl  is  the  internal  lining  of  the  shell  of  a  species  of  oyster.  It  is  composed  of  extremely 
thin  laminae  of  a  tough  and  elastic,  yet  at  the  same  time  hard  and  shelly  substance,  disposed  parallel  to  the 
irregular  concavity  of  the  interior  of  the  shell.  When,  therefore,  any  portion  of  it  is  ground  and  polished  on  a 
plane  tool,  the  artificial  surface  so  produced  intersects  the  natural  surfaces  of  the  laminae  in  a  series  of  undulating 
curves,  or  level-lines,  which  are  nearer  or  farther  asunder,  according  to  the  varying  obliquity  of  the  artificial  to 
the  natural  surfaces.  As  these  laminae  adhere  imperfectly  to  each  other,  their  feather-edges  become  broken  up 
by  the  action  of  the  powders,  &c.  used  in  grinding  and  polishing  them,  so  as  to  present  a  series  of  ridges  or 
escarpments  arranged  (when  any  very  small  portion  of  the  surface  only  is  considered)  nearly  parallel  to,  and 
equidistant  from  each  other,  which  are  distinctly  seen  with  a  microscope,  and  which  no  polishing  in  the  least 
degree  obliterates  or  impairs.  The  light  reflected,  therefore,  or  dispersed  on  their  edges,  will  interfere  and 
produce  coloured  appearances  in  a  direction  perpendicular  to  that  of  the  stria?.  This  is,  in  fact,  their  situation  ; 
but  the  phenomena  are  modified  in  a  very  singular  manner  by  the  peculiar  form  of  the  edges  and  hollows, 
which  results,  no  doubt,  from  the  crystalline  structure  of  the  pearl.  That  it  is  the  configuration  only  of  the 
surface  on  which  they  depend,  is  evident  from  the  remarkable  fact,  that,  like  the  colours  described  in  Art.  702, 
they  may  be  transferred,  by  impression,  to  sealing  wax,  gum,  resin,  or  even  metals,  with  little  or  no  diminution 
of  their  brilliancy ;  and  the  impression  so  transferred,  if  examined  by  the  microscope,  is  found  to  exhibit  a 
faithful  copy  of  the  original  striae,  though  sometimes  so  minute  as  hardly  to  exceed  one  3000th  of  an  inch  in 
their  distance  from  each  other.  For  a  particular  description  of  this  very  curious  and  beautiful  class  of  pheno- 
mena, however,  our  limits  oblige  us  to  refer  to  the  original  memoirs  already  cited,  especially  as  their  theory  is 
still  accompanied  with  some  obscurity. 

§  VIII.     Of  the  Diffraction  of  Light. 

When  an  object  is  placed  in  a  very  small  beam  of  light,  or  in  the  cone  of  rays  diverging  from  an  extremely      706. 
small  point,  such  as  a  sunbeam  admitted  through  a  small  pin-hole  into  a  dark  chamber,  or,  still  better,  through  Fringes 
an  opening  of  greater  size,  behind  which  a  lens  of  short  focus  is  placed,  so  as  to  form  an  extremely  minute  and  fnrmed  «"- 

brilliant  image  of  the  sun  from  which  the  rays  diverge  in  all  directions,  its  shadow  is   observed  to  be  bordered  't"0/  :°  "'," 

i,  e       i  ,   c  •  ,  .  «          shadows  of 

externally   by  a  series  ot  coloured  fringes  which  are  more  distinct  the    smaller  the  angular  diameter  of  the  bodies  in  a 

luminous  point,  as  seen  from   the  object.     If  this  be  much  increased,   the   shadow  and  fringes  formed  by  its  small  beam 
several  points,  regarded  each  as  an  independent  luminary,  overlap  and  confuse  each  other,   obliterating  the  °f  ''gnt 
colours,  and  producing  what  is  called  the  penumbra  of  the  object ;  but  when  the  luminous  point  is  extremely 
minute,  the  shadow  is  comparatively  sharp,  and  the  fringes  extremely  well  defined. 

These  fringes  (which  were  first  described  by  Fattier  Grimaldi  in  a  work  entitled  Physico-Mathesis  de  Lumine,  707. 
Bologna,  1665,  and  afterwards  more  minutely  by  Newton  in  the  third  book  of  his  Optics)  surround  the  shadows  of Tlleir 
objects  of  all  figures,  preserving  the  same  distance  from  every  part,  like  the  lines  along  the  sea-coast  in  a  map ; 
only,  where  the  object  forms  an  acute,  salient  angle,  the  fringes  curve  round  it ;  and  where  it  makes  a  sharp, 
reentering  one  they  cross,  and  are  carried  up  to  the  shadow  at  each  side,  without  interfering  or  obliterating  each 
other.  In  white  light  three  only  are  to  be  seen,  whose  colours,  reckoning  from  the  shadow,  are  black,  violet, 
deep  blue,  light  blue,  green,  yellow,  red ;  blue,  yellow,  red  ;  pale  blue,  pale  yellow,  pale  red.  In  homogeneous 
light  they  are,  however,  more  numerous,  and  of  different  breadths,  according  to  the  colours  of  the  light,  being 
narrowest  in  violet  and  broadest  in  red  light,  as  in  the  coloured  rings  between  glasses ;  and  it  is  by  the  mutual 
superposition  of  the  different  sets  of  fringes  for  all  the  coloured  rays  that  their  tints  are  produced,  and  their 
obliteration  after  a  few  of  the  first  orders  caused. 

The  fringes  in  question  are  absolutely  independent  of  the  nature  of  the  body  whose  shadow  they  surround,      708. 
and  the  form  of  its  edge.     Neither  the  density  or  rarity  of  the  one,  nor  the  sharpness  or  curvature  of  the  other,  Are  i"de 
having  the  least  influence  on  their  breadth,  their  colours,  or  their  distance  from  the  shadow  ;  thus  it  is  indifferent  Pen^e "'  ot 
whether  they  are  formed  by  the  edge  or  back  of  a  razor,  by  a  mass  of  platina  or  by  a  bubble  in  a  plate  of  glass,  Ca5tin°s  the 
(which,  though  transparent,  yet  throws  a  shadow  by  dispersing  away  the  light  incident  on  it,)  circumstances  shadow, 
which  make  it  clear  that  their  origin  has  no  connection  with  the  ordinary  refractive  powers  of  bodies,  or  with 
any  elective  attraction  or  repulsions  exerted  by  them  on  light ;  for  such  forces  cannot  be  conceived  as  independent 
of  the  density  of  the  body  exerting  them,  however  minute  we  might  regard  the  sphere  of  their  action. 

To  see  the  fringes  in  question,  they  may  be  received  on  a  smooth,  white  surface,  and  examined  and  measured      709. 
thereon  by  contrivances  which  readily  occur ;  this  was  the  mode  pursued  by  Newton.     M.  Fresnel,  however,  M-  Fresnel  a 
having  (to  avoid  the  inconvenience  of  intercepting  the  light  by  the  interposition  of  the  observer)  received  them  on  an  IMt^?i0' 
emeried  glass  plate,  was  enabled,  by  placing  himself  behind  it,  to  approach  uear  enough  to  examine  and  measure  them"" 


480 


LIGHT. 


Light. 


710. 

Their  phe- 
nomena. 

1st  Their 
distances, 
inter  se. 
711. 
They  are 
propagated 
in  curved 
lines. 


Fig.  142. 


712. 

The  visible 
snadow 
differs  from 
the  geome- 
trical one 
mnd  is  larger. 


713. 

Newton's 
doctrine  of 
the  deflex- 
ion of  light. 
Fig.  143. 


714. 

His  account 
of  the 
fringes. 
Fig.  144. 


715. 

Newton's 

doctrine 

and 

Kresnel's 

objections 

to  it  con- 

lidered. 


them  with  a  magnifier.     In  so  doing,  however,  he  observed,  that  when  thus  once  brought  under  inspection,  they    Part  III. 
continued  visible,  and  were  indeed  much  brighter  and  more  distinct  in  the  focus  of  the  lens  (as  if  depicted  in  the  >^-^— 
air)  even  when  the  emeried  glass  was  altogether  withdrawn ;  and  this  fortunate  observation,  by  enabling  him  to 
avoid  the  use  of  a  screen  altogether,  and  to  perform  all  his  measurements  of  their  dimensions  by  the  aid  of  a 
micrometer,  put  it  in  his  power  to  examine  them  with  a  degree  of  minuteness  and  precision  no  other  way  attain- 
able, and  fully  adequate  to  the  delicacy  of  the  inquiry :  for  it  is  manifest  that  the  fringes,  being  seen  as  they 
would  be  formed  if  received  on  a  screen  in  the  focus,  may  be  regarded  as  any  other  optical  image  formed  in  the 
focus  of  a  telescope,  viewed  with  any  magnifier,  and  treated  in  all  respects  as  such  images. 

Whatever  mode  of  examining  them  we  adopt,  however,  we  shall  observe  the  following  facts : 

Phenomenon  1.  That,  creteris  paribus,  the  distances  from  each  other  and  from  the  border  of  the  shadow 
diminishes  as  the  screen  on  which  they  are  received,  or  the  plane  in  the  focus  of  the  lens  in  which  they  are 
formed,  approaches  the  border  of  the  opaque  body,  and  ultimately  coincides  with  it,  so  that  they  seem  to  have 
their  origin  close  to  the  edge  of  the  body. 

Phenomenon  2.  That  they  are  not,  however,  propagated  in  straight  lines  from  the  edge  of  that  body  to  a 
distance,  but  in  hyperbolic  curves,  having  their  vertices  at  that  edge  ;  and  therefore  that  it  is  not  one  and  the 
same  light  which  forms  one  and  the  same  fringe  at  all  distances  from  the  opaque  body.  To  explain  this, 
conceive  the  distances  of  the  fringes  from  each  other  and  from  the  shadow  measured  accurately  at  a  great  variety 
of  distances  from  the  edge  of  the  body  ;  then,  were  they  propagated  in  straight  lines,  and  were  each  fringe  really 
the  axis  of  a  pencil  of  rays  emanating  from  a  point  at  that  edge,  their  intervals  and  distances  from  the  shadow 
ought  to  be  proportional  to  the  distances  from  the  edge  of  the  body ;  but  it  is  not  so,  in  fact, — the  former 
distances  increasing  as  we  recede  from  the  opaque  body  much  more  rapidly  at  first,  and  less  so  as  we  recede, 
than  according  to  the  law  of  proportionality  ;  and  if  the  locus  of  each  fringe  be  laid  down  from  such  measures, 
it  will  be  found  to  be  an  hyperbolic  curve  having  its  convexity  outwards  or  from  the  shadow.  Thus  in  fig.  142 
O  is  the  luminous  point,  A  the  edge  of  the  body,  and  G  H  a  screen  perpendicular  to  the  straight  line  O  A,  C 
the  border  of  the  visible  shadow,  and  D,  E,  F  the  places  of  the  successive  minima  of  the  fringes  in  a  line  at 
right  angles  to  the  edge  of  the  shadow.  If  the  screen  be  brought  nearer  to  the  body  A  as  at  gh,  and  if  c,  d,  e,f 
be  the  points  corresponding  to  C  D  E  F,  their  loci  will  be  the  hyperbolas  AcC,  A  d  D,  &c. 

It  will  be  noticed  also  that  the  border  C  of  the  visible  shadow  is  not  coincident  with  B,  that  of  the  geometrical 
one,  which  lies  in  the  straight  line  O  A,  grazing  the  edge  of  the  object.  The  deviation  is  difficult  to  perceive  in 
the  shadow  of  a  large  body,  having  nothing  to  measure  from ;  but  if  we  examine  those  of  very  narrow  bodies, 
as  of  a  hair,  for  instance,  in  such  a  beam  of  light  as  described,  we  shall  find  on  measuring  the  total  breadth  of 
the  shadow  a  full  proof  of  this.  This  fact  was  observed  by  Grimaldi.  The  limit  of  the  visible  shadow  also 
follows  the  same  law  of  curvilinear  propagation  as  the  fringes.  Thus,  Newtou  found  the  shadow  of  a  hair 
one  280th  of  an  inch  in  diameter  placed  at  12  feet  distance  from  the  luminous  point,  to  measure  at  4  inches 
from  the  hair  ^v  inch,  or  upwards  of  4  diameters  of  the  hair,  at  two  feet,  -£f  inch,  or  10  diameters;  while  at  10 
feet  it  measured  only  -1  inch,  or  35  diameters,  instead  of  120,  which  it  should  have  done  if  the  rays  terminating 
the  shadow  had  proceeded  in  straight  lines ;  or  rather,  to  speak  more  correctly,  if  the  shadow  were  bounded  by 
straight  lines. 

To  account  for  these  remarkable  facts,  Newton  supposes  that  the  rays  passing  at  different  distances  from  the 
edges  of  bodies  are  turned  aside  outwards,  as  if  by  a  repulsive  force ;  and  that  those  nearest  are  turned  more 
aside  than  those  more  remote,  as  in  fig.  143,  where  X  is  a  section  of  the  hair,  and  AD,  BE,  CF,  &c.  rays 
which  pass  at  different  distances  beside  it,  and  which  are  turned  off  at  angles  rapidly  diminishing  as  the  distance 
increases  in  directions  D  G,  E  H,  F  I,  &c.  It  is  manifest  that  the  curve  W  Y  Z,  to  which  all  these  deflected 
rays  are  tangents,  and  within  which  none  can  enter,  will  be  convex  outwards ;  and  its  curvature  will  be  greatest 
at  the  vertex  W,  and  will  diminish  continually  as  it  recedes  from  X,  being,  in  fact,  the  caustic  of  all  the 
deflected  rays. 

This  will  be  the  boundary  of  the  visible  shadow.  To  account  for  the  fringes,  he  supposes  (Optics,  book  iii. 
question  3)  that  each  ray  in  its  passage  by  the  body  undergoes  several  flexures  to  and  fro,  as  in  fig.  144  at  a, 
b,  c  ;  and  that  the  luminous  molecules,  of  which  that  ray  consists,  are  thrown  off  at  one  or  other  of  the  points 
of  contrary  flexure,  or  other  determinate  points  of  the  serpentine  curve  described  by  them  according  to  the 
state  of  their  fits  in  which  they  there  happen  to  be,  or  other  circumstances ;  some  outwards,  as  in  the  directions 
a  A,  6B,  cC,  dD,  and  others  we  may  suppose  inwards,  as  a  a,  6/3,  07,  &c.  With  the  latter  we  have  here 
no  concern.  The  former,  it  is  evident,  will  give  rise  to  as  many  such  caustics  as  above  described,  as  there 
are  deflected  rays;  and  each  caustic,  when  intercepted  on  a  screen  at  a  distance,  will  depict  on  it  the  maximum 
of  a  fringe.  The  intervals,  however,  between  these  caustics,  or  minima  of  the  fringes,  will  not  be  totally  black ; 
because  the  rays  from  the  other  caustics,  after  crossing  on  the  confines  of  the  shadow,  or  interior  fringes,  wi'.l 
pursue  their  course,  and  partially  illuminate  all  the  space  beyond.  Thus  the  fringes  should  be  less  numerous 
and  the  degradation  of  colour  more  rapid  than  in  the  coloured  rings. 

This  theory  accounts  then  perfectly  for  the  curvilinear  propagation  of  the  fringes,  for  their  rapid  degradation, 
for  their  apparently  originating  in  the  very  edge  of  the  body,  (since  each  caustic  will  actually  come  up  to  that 
edge,  as  at  A,  fig.  142,)  and  for  the  remarkable  brightness  of  the  fringes,  especially  the  first,  which  really 
contains  in  itself  all  the  light  which  would  have  passed  into  the  region  B  C  between  the  visible  and  geome- 
trical shadows.  It  should  appear,  therefore,  that  M.  Fresnel,  in  the  objections  he  has  taken  against  these  points 
of  the  Newtonian  doctrine  of  inflexion  in  his  excellent  work  Sur  la  Diffraction  de  la  Lumiere,  (§  1,  p.  15,  17,  19,) 
must  have  formed  a  very  inadequate  conception  of  the  doctrine  he  opposes,  which,  if  viewed  in  the  light  he  has 
there  placed  it  in,  would  indeed  deserve  no  other  epithet  than  puerile,  and  must  be  looked  upon  as  quite  unworthy 
of  its  illustrious  author ;  and  were  these  the  only  difficulties  to  be  explained,  we  should  certainly  not  be  justified 


LIGHT.  481 

in  passing  a  hasty  sentence  on  it.     Other  objections  advanced  by  the  same  eminent  philosopher,  however,  are    Part  III. 
'  more  serious,  and  refer  to  a  phenomenon  of  which  the  doctrine  of  deflective  forces  seems  incapable  of  giving  <—  -  v"—- 
any  account  ;    but  of  which,  in  justice  to  Newton  we  ought  to  add,  it  does  not  appear  that  he  was  aware,  or 
its  importance  could  not  fail  to  have  struck  him. 

Phenomenon  3.  All  other  things  remaining  the  same,  let  the  opaque  body  A  be  brought  nearer  the  luminous      716. 
point  O,  (fig.  142.)    The  fringes  then,  formed  at  the  same  distance  as  before  behind  A,  are  observed  to  dilate  con-  Dilatation 
siderably  in  breadth,  —  preserving,  however,  the  same  relative  distances  from  each  other,  and  from  the  border  of  the  ^  tlle 
shadow.     This  fact  is  evidently  incompatible  with  the  idea  of  their  being  caused  by  any  deflecting  force  emanating  |^"|es  by 
from  the  opaque  body,  since  it  is  inconceivable  that  such  a  force  should  depend  on  the  distance  the  light  has  p"oach"of 
travelled  from  another  point  no  way  related  to  the  body.  the  radiant 

To  explain  the  diffracted  fringes  on  the  undulatory  doctrine,  Dr.  Young  conceived  the  rays  passing  near  the  Point- 
edge  of  the  opaque  body  to  interfere  with  those  reflected  very  obliquely  on  its  edge,  and  which  in  the  act  of      717. 
reflexion  had  lost  half  an  undulation,  as  in  the  case  of  the   reflected  rings.     This  supposition  would,  in  fact,  ^-Young's 
lead  us  to  conclude  the  existence  of  a  series  of  fringes  propagated  hyperbolically,   and  perfectly  resembling  oHhe"1 
those  really  existing.     M.  Fresnel,  however,  has  shown  that  a  minute  though  decided  difference  exists  between  fringes  on 
their  places,  as  given  by  this  theory  and  by  direct  measurement  ;   and  has,  moreover,  remarked,  that  were  this  the  undula- 
the  true  explanation,  they  could  hardly  be  supposed  absolutely  independent  of  the  figure  of  the  edge  of  the  'ory  . 
opaque  body,  which  experience  shows  they  are  ;   and  that  in  cases  where  this  edge  is  extremely  sharp,  the  small  o^*e"Jio'ns 
quantity  of  light  which  could  be  reflected  from  it  would  be  insufficient  to  interfere  with  that  passing  by  it,  so  as  against"!?. 
to  form  fringes  so  bright  as  we  see  them.     These  objections  appear  conclusive,  especially  as  the  supposition  of 
a  reflexion  on  the  edge  of  the  body  is  unnecessary,  since  a  more  strict  application  of  the  undulatory  doctrine, 
assisted  by  the  principle  of  interferences,  will  be  found  to  afford  a  full  and  precise  explanation  of  all  the  facts, 
regarding  the  opaque  body  as  merely  an  obstacle  bounding  the  waves  propagated  from  the  luminous  point  on 
one  side. 

To  show  this,  let  us  consider  a  wave  AMP   propagated  from  O,  and  of  which  all  that  part  to  the  right  of  A      718. 
(fig.  145)  is  intercepted  by  the  opaque  body  A  G  ;  and  let  us  consider  a  point  P  in  a  screen  at  the  distance  A  B  Fresnel's 
behind  A,  as  illuminated  by  the  undulations  emanating  simultaneously  from  every  point  of  the  portion  A  M  F,  explanation 
according  to  the  theory  laid  down  in  Art.  628,  et  seq.     For  simplicity,  let  us  consider  only  the  propagation  of  F'°'  145- 
undulations  in  one  plane.     Put  AO  =  a,  AB  =  6,  and  suppose  \—  the  length  of  an  undulation;  and  drawing 
P  N  any  line  from  P  to  a  point  near  M,  put  P  F  =/i  NM  =  s,  PB  =  x;  then,  supposing  P  very  near  to  B, 

and  with  centre  P  radius  P  M  describing  the  circle  Q  M,  we  shall  have  /=  PQ-j-QN=  ^  (a  -f"  ft)*  +  **  —  a 

•4-  Q  N  =  6  -I  --  ;  ---  1-  Q  N.    Now,  Q  N  is  the  sum  of  the  versed  sines  of  the  arc  s  to  radii  O  M  and  P  M, 
2  (a  +  b) 

and  is  therefore  equal  to  ^  +  ^  =   ~(~  +  I)  =  yjL*  .  f  .   so  that,  finally, 

/=6+         **         +     0*  +  ^°. 
J  2  (a+b)  2  a  6 

Now,  if  we  recur  to  the  general  expression  demonstrated  in  Art.  632,  for  the  motion  propagated  to  P  from 
any  limited  portion  of  a  wave,  we  shall  have  in  this  case  o  .  0  (<?)  =  1,  because  we  may  regard  the  obliquity  of 
all  the  undulations  from  the  whole  of  the  efficacious  part  of  the  surface  A  M  N  as  very  trifling,  when  P  is  very 
distant  from  A  in  comparison  with  the  length  of  an  undulation.  And  as  we  are  now  only  considering  undu- 

/  t          f\ 

lations  propagated  in  one  plane,  that  expression  becomes  merely  V  =  J~ds  .  sin  2  ir  [  —  —  —  J,  and  the  cor- 

responding expression  for  the  excursions  of  a  vibrating  molecule  at  P  will  be 

X  =/<£*.  cos  2 
If  then  we  put  for  f  its  value,  and  take 


(t         b  a«        \  /2(«  +  6) 

2  *  VT  -  x  ~  2T^+T))-  '•  V      «6x     -  "• 

and  consider  that  in  those  expressions  t  and  x  remain  constant,  while  s  only  varies,  the  latter  will  take  the 
form 


_ 

=  v  yj^y  •  {cos  °  -fd  "-cos  (ir  "*)  +  sin  e  -fd  "  •  si 


which  shows  that  the  total  wave  on  arriving  at  P  may  be  regarded  as  the  resultant  of  two  waves  X'  .  cos  0 
and  X"  .  sin  0,  differing  in  their  origin  by  a  quarter-undulation,  and  whose  amplitudes  X'  and  X"  are  given  by 
the  expression 


.   /      a 

=  v 


b\  r,  f  .  ab\ 

d''cos''°;     x  :: 


the  integrals  being  taken  between  limits  of  v  corresponding  to  s  e=  —  A  M,  and  $  =  -f-  cc.     Consequently, 
VOL.  iv.  3  R 


482 


LIGHT. 


1  since 
the  limits  of  v  must  be 


=  AM=PBx 


a  -f-  b       a  + 


r,  and  v  = 


.   / 
W 


2  (a  4-  6) 

•        .T  -, 

a  6  X 


Part  III. 


=  -  x  V/  7  —  ,    ^  -  V  and  "  = 
v     («  -f-  o)  6  X 


719. 

Rule  for 
determining 
the  illumi- 
nation of 
any  point  in 
the  screen. 

720. 
Maxima 
and  minima 
numerically 
estimated. 


Hence,  to  determine  the  intensity  of  the  light  at  any  point  P  on  the  screen,  we  must  first  of  nil  calculate 
the  values  of  these  integrals  ;  and  having  thus  determined  X'  and  X",  the  square  root  of  the  sum  of  their  squares 

V  X'2  -f-  X"2  will  represent  the  amplitude  of  a  single  vibration,  the  resultant  of  hoth,  (Art.  615  ;)  and  the  sum  of 
their  squares  simply  (X'2  -f-  X"2),  the  intensity  of  the  light,  or  the  sensation  produced  in  the  eye. 

M.  Fresnel,  in  the  work  already  cited,  has  given  a  table  of  the  values  of  these  integrals  for  limits  succes- 
sively increasing  from  0  up  to  ce,  (at  which  latter  limit  each  is  equal  to  £,  as  may  readily  be  proved  ;)  and,  calcu- 
lating on  this,  he  finds  that  the  intensity  of  the  light,  without  the  limit  of  the  geometrical  shadow,  passes 
through  a  series  of  maxima  and  minima  according  to  the  following  table  : 

Table  of  the  Maxima  and  Minima  for  the  Exterior  Fringes,  and  of  the  Corresponding  Intensities  of  the  Light 

illuminating  them. 


Values  of  ». 

Intensities 
of  the  light. 

Values  of  ». 

Intensities 
of  the  light. 

First  maximum  .... 

1.2172 

2.7413 

Fourth  minimum    .  . 

3.9372 

1.7783 

First  minimum  .... 

1.8726 

1.5570 

Fifth  maximum  .... 

4.1832 

2.2206 

Second  maximum  .  . 

2.3449 

2.3990 

Fifth  minimum  .... 

4.4160 

1.8014 

Second  minimum   .  . 

2.7392 

1.6867 

Sixth  maximum  .... 

4.6069 

2.1985 

Third  maximum 

3.0820 

2.3022 

Sixth  minimum  .... 

4.8479 

1.8185 

Third  minimum  .... 

3.3913 

1.7440 

Seventh  maximum.  . 

5.0500 

2.1818 

Fourth  maximum  .  . 

3.6742 

2.2523 

Seventh  minimum  .  . 

5.2442 

1.8317 

721. 

Illumina- 
tion of  the 
border  of 
the  geome- 
trical 
shadow. 
722. 

Illumina- 
tion within 
the  shadow. 


In  this  it  is  to  be  remarked,  that  no  minimum  is  zero,  and  that  the  difference  between  the  successive  maxima 
and  minima  diminishes  very  rapidly  as  the  values  of  v  increase,  which  explains  the  rapid  degradation  of  their 
tints. 

If  the  point  P  be  situated  on  the  very  edge  of  the  geometrical  shadow,  its  illumination  should  on  this  theory 
be  (i  )2  +  (£)2  =  \.  To  compare  this  with  the  illumination  of  the  same  point,  were  the  opaque  body  removed, 
we  have  only  to  consider,  that  at  a  great  distance  from  the  shadow  the  light  must  be  the  same,  whether  the 
body  be  there  or  not.  Now  the  limit  to  which  the  maxima  ]and  minima  approximate  is  2,  which  therefore 
represents  the  uniform  illumination  beyond  the  fringes  ;  so'  that  the  light  on  the  border  of  the  geometrical 
shadow  is  equal  to  \  of  the  full  illumination  from  the  radiant  point. 

Within  the  shadow  we  have  only  to  make  s  or  i  negative.     This  does  not  alter  the  values  of  the  integrals, 


but  it  does  their  limits,  which  must  in  that  case  be  taken  not  from  v  =  — 


2a 


to  -f-  CD,  but 


from  v  = 


2a 


;  +  6)  6  X 
to  -(-co.     The  computations   have  been   executed  by  M.  Fresnel,  who  finds 


723. 

Visible 
shadow 
larger    than 
•he  geome- 

•riral. 

724. 


(o  +  6)  6  X 

that  no  periodical  increase  or  decrease  here  takes  place,  but  that  the  light  degrades  rapidly  and  constantly 
within  the  geometrical  shadow  to  total  darkness. 

The  actual  visible  shadow  then  is  marked  by  no  sudden  defalcation  of  light,  and  it  will  depend  on  the  judgment 
of  the  eye  where  to  establish  its  termination.  If  we  regard  all  that  part  as  shadow  which  is  less  illuminated  than 
the  general  light  of  the  screen  beyond  the  fringes,  then  the  visible  shadow  will  extend  considerably  beyond  the 
geometrical  one,  and  this  explains  why  the  shadows  of  small  bodies  are  so  much  dilated,  as  we  have  seen 
they  are. 

If  we  would  know  the  breadths  of  the  several  fringes,  we  have  only  to  find  the  values  of  x  in  the  equation 


x  = 


(a  +  6)  6 


where  v  has  in  succession  the  several  values  set  down  in  the  foregoing  table.  If  we  consider  the  variation  of  x 
for  successive  values  of  a  and  6,  we  shall  see  the  origin  both  of  the  curvilinear  propagation  of  the  fringes,  and  of 
their  dilatation  on  the  approach  of  the  luminous  point.  In  fact  if  we  regard,  first,  the  relation  between  b  and  x, 
or  the  locus  of  any  fringe  regarded  as  a  curve,  having  the  line  A  B  for  an  abscissa  and  B  P  as  an  ordinate,  we 

have  x*  =  v"  — -I  5  H Y  which  is  the  equation  of  an  hyperbola  having  its  convexity  outwards  and  passing 

*  \  a  / 

through  A.  Secondly,  on  the  other  hand,  if  we  regard  a  as  the  variable  quantity  and  b  as  constant,  we  see  that  for 
one  and  the  same  distance  from  the  screen,  the  breadths  of  the  fringes  increase  as  a  diminishes ;  the  increments  of 


LIGHT.  483 

their  squares,  as  the  incident  rays  from  being   parallel  become  more  divergent,  being  directly  as  their  diver-   P.n-i  Ilj. 
s  gence.     Thirdly,  for   equal  values  of  X,  a,  and  A,  x  is  proportional  to  v ;  so  that  the  breadths  of  the  several  v_-v-»^ 
fringes  are  always  in  the  same  ratio  to  each  other,  and  form  a  progression  the  same  with  those  of  the  values  of 
v  in  the  foregoing  table.    Lastly,  the  breadths  of  the  fringes  for  different  coloured  rays  are  as  the  square  roots  of 
the  lengths  of  their  undulations. 

The  accordance  of  this  theory  with  experiment,  so  far  as  it  regards  the  distances  of  the  fringes  from  the  725. 
shadow  and  from  each  other,  has  been  put  to  a  severe  test  by  M.  Fresnel,  and  found  perfect.  It  were  to  be 
wished,  however,  that  he  had  stated  somewhat  more  precisely  the  instrumental  means  by  which  he  determined 
the  place  of  the  border  of  the  geometrical  shadow,  from  which  his  measures  are  all  stated  to  be  taken ;  and 
which,  being  marked  by  no  phenomenon  of  maximum  or  minimum,  might  be  liable  to  uncertainty  if  judged  of 
by  the  eye  alone.  This,  however,  in  no  way  invalidates  the  accuracy  of  the  final  conclusions,  as  the  intervals 
between  the  fringes  are  distinctly  marked,  and  susceptible  of  exact  measurement.  The  dilatation  of  the  fringes 
on  the  approach  of  the  luminous  point  is,  perhaps,  the  strongest  fact  in  favour  of  the  undulatory  doctrine,  and 
in  opposition  to  that  of  inflection,  which  has  yet  been  adduced.  It  seems  hardly  reconcilable  to  any  received 
ideas  of  the  action  of  corpuscular  forces,  to  suppose  the  force  of  deflection  exerted  by  the  edge  of  a  body- 
on  a  passing  ray,  to  depend  on  the  distance  which  the  ray  has  passed  over  before  arriving  at  that  edge  from 
an  arbitrarily  assumed  origin.  M.  Fresnel  has  placed  this  argument  in  a  strong  light,  in  his  work  already  cited. 

Besides  the  exterior  fringes  above  described,  there  are  others  formed  in  certain   circumstances   within  the      726. 
shadows  of  bodies  which  afford  peculiarly  apt  illustrations  of  the  principle  of  interferences.     The  first  class  of  Fringes. 
phenomena  of  this  kind  was  noticed  by  Grimaldi,  who  found  that  when  a  long,  narrow  body  is  held  in  a  small  observed  I  v 
diverging  beam  of  light,  the  shadow  received  on  a  screen  at  a  distance  will  be  marked  in  the  direction  of  its  tinmal'il 
length  with   alternate  streaks  or  fringes  brighter  and  darker  than  the  rest.     These  are  more  or  less  numerous,  "arrow 
according  as  the  distance  of  the  shadow  from  the  body  is  smaller  or  greater  in  proportion  to  the  breadth  of  the  shadows. 
latter.     To   study  the  phenomena   more  minutely,  Dr.  Young  passed  a  sunbeam  through  a  hole  made  with  a 
fine  needle  in  thick  paper,  and  brought  into  the  diverging  beam  a  slip  of  card  one-thirtieth  of  an  inch  in  breadth, 
and  observed  its  shadow  on  a  white  screen  at  different  distances.     The  shadow  was  divided  by  parallel  bands, 
as  above  described,  but  the  central  line  was  always  white.     That  these  bands  originated  in  the  interference  of  Dr. Young's 
the  light   passing  on  both  sides  of  the  card,  Dr.  Young  demonstrated  beyond    all  controversy,   by  simply  flln<lan|cn- 
intercepting  the  light  on  one  side  by  a  screen  interposed  between  the  card  and  the  shadow,  leaving  the  rays  f^" 
on  the  other  side  to  pass  freely,  in  the  manner  represented  in  fig.  146,  where   O  is    the   hole,  A  B   the   card,  y\g,  )  in 
E  F  its  shadow,  and  C  D  the  intercepting  body  receiving  on  its  margin  the  margin  of  the  shadow  of  the  edge 
B  of  the  body.      In  this  arrangement  all  the  fringes  which  had  before  uxisted  in  the  shadow  E  F  immediately 
disappeared,  although  the  light  inflected    on  the    edge  A   was  allowed  to   retain  its  course,  and  must  have 
necessarily   undergone  any  modification  it  was  capable  of  receiving  from  the  proximity  of  the  other  edge  B. 
The  same  result  took  place  when  the   intercepting  screen  was  placed  as  at  c  d  before  the  edge  B  of  the  body, 
so  as  to  throw  its  own  shadow  on  the  margin  B  of  the  card. 

Without  entering  minutely  into  the  rationale  of  this  phenomenon,  which,  however,  the  formulae  of  the  pre-       727. 
ceding  articles  enable  us  fully  to  do,  by  considering  the  illumination    of  any  point  X   between  E  and  F  as  Expla- 
arising  from  the  whole  wave  a  A  B  6,  minus  the  portion  A  B,  and  which  M.  Fresnel  has  done  at  full  length,  nation. 
and  with  great  success,  in  his  Memoir  already  so  often  cited  ;    we  shall  content  ourselves  with  showing  how 
fringes  or  alternations  of  colour  must  originate  in  such   circumstances;    in   fact,   if  we  join  AX,  B  X,  the 
difference  of  routes  of  the  waves  arriving  at  X   by  the  paths  O  AX,  O  B  X  is   equal  to  B  X  —  A  X.     It  is 
therefore  nothing  in  the  middle  of  the  shadow  E  F,  which  ought  therefore  to  be  illuminated  by  double  the  light 
deflected  into  the  shadow  at  that  distance  by  either  edge,  Art.  722,  which  will  be  less  as  the  object  is  larger, 
and  the  shadow  broader.     But  on  either  side  of  the  middle  B  X  —  A  X  increases ;    and  when  it  attains  a  value 
equal  to  half  an  undulation,  the  waves  are  in  complete  discordance,  and  therefore  the  middle  bright  portion  will 
be  succeeded  by  a  dark  band  on  either  side,  and  these  again  by  bright  ones,  and  so  on. 

An  elegant  variation  of  this  experiment  of  Dr.  Young  is  afforded  by  a  phenomenon  described  by  Grimaldi.       728. 
When  a  shadow  is  formed  by  an  object  having  a  rectangular  termination  ;    besides  the  usual  external  fringes  GrinnMi  s 
there  are  two  or  three  alternations  of  colours,  beginning  from  the  line  which  bisects  the  angle,  disposed,  within  cre •'' ''' 
the  shadow  on  each  side  of  it,  in  curves  which  are  convex  towards  the  bisecting  line,  and  which  converge  towards  'nnSes- 
it  as  they  become  remote  from  the  angular  point.     These  fringes  are  the  joint  effect  of  the  light  spreading  into 
the  shadow  from  each  outline  of  the  object,  and  interfering  as  above  ;  and  that  they  are  so,  is  proved  by  placing 
a  screen  within  a  few  inches  of  the  object,  so  as  to  receive  only  one  edge  of  the  shadow,  when  the  whole  of  the 
fringes  disappear.     If,  on  the  other  hand,  the  rectangular  point  of  the  screen  be  opposed  to  the  point  of  the 
shadow,  so  as  barely  to  receive  the  angle  of  the  shadow  on  its  extremity,  the  fringes  will  remain  undisturbed. 
(Young,  Experiments  and  Calculations  relating  to  Physical  Optics,  Phil.  Trans.,  1803.) 

Such  are  some  of  the  more  remarkable  appearances  produced  within  and  beyond  the  shadows  of  narrow       729. 
bodies.     Let  us  next  consider  the  effect  of  transmitting  a  beam  through  a  very  narrow  aperture.     And  the  first-Ca?e  °^ 
case  is  when  the  aperture  is  circular.     Suppose,  for  instance,  we  place  a  sheet  of  lead,  having  a  small  pin-hole  ll 
pierced  through  it,  in  the  diverging  cone  of  rays  from  the  image  of  the  sun,  formed  by  a  lens  of  short  focus,  and  mall  ' 
in  the  line  joining  the  centres  of  the  hole  and  focus  prolonged  place  a  convex  lens  or  eye-glass,  behind  which  circular 
the  eye  is  applied.     The  image  of  the  hole  will  be  seen  through  the  lens  as  a  brilliant  spot,  encircled  by  rings  aperture, 
of  colours  of  great  vividness,  which  contract  and  dilate,  and  undergo  a  singular  and  beautiful  alternation  of  tints, 
as  the  distance  of  the  hole   from   the  luminous  point  on  the  one  hand,  or  on  the  eye-glass  on  the  other,  is 
changed.     When  the  latter  distance  is  considerable,  the  central   spot  is  white,  and  the  rings  follow  nearly  the 
order  of  the  colours  of  thin  plates.     Thus,  when  the  diameter  of  the  hole  was  about  T'ath  of  an  inch,  its  distance 

3  a  2 


484 


LIGHT. 


light,      («)  from  the  luminous  point  about  6  feet  6  inches,  and  its  distance  (6)  from  the  eye-lens  24  inches,  the  series   Part  HI. 
»^»^  of  colours  was  observed  to  be,  ^ ^v^*1 

1st  order.     White;  pale  yellow  ;  yellow;  orange;  dull  red. 

Violet ;  blue  (broad  and  pure ;)  whitish  ;  greenish  yellow  ;  fine  yellow ;  orange  red,  very  full  and 


730. 

Table  of 
colours  of 


2d  order, 
brilliant. 
3rd  order. 
4th  order. 
5th  order. 
6th  order. 
7th  order. 


Purple;  indigo  blue;  greenish  blue ;  pure,  brilliant  green  ;  yellow  green  ;  red. 
Good  green,  but  rather  sombre  and  bluish  ;  bluish  white  ;  red. 
Dull  green  ;  faint  bluish  white  ;  faint  red. 
Very  faint  green  ;  very  faint  red. 
A  trace  of  green  and  red. 

When  the  eye-lens  and  hole  are  brought  nearer  together,  the  central  white  spot  contracts  into  a  point  and 
vanishes,  and  the  rings  gradually  close  in  upon  it  in  succession,  so  that  the  centre  assumes  in  succession  the 
I  most  surprisingly  vivid  and  intense  hues.  Meanwhile  the  rings  surrounding  it  undergo  great  and  abrupt  changes 
and  sur-P°  'n  'h6"  t'"18-  T'le  following  were  the  tints  observed  in  an  experiment  made  some  years  ago,  (July  12,  1819,) 
rounding  the  distance  between  the  eye-glass  and  luminous  point  («  +  6)  remaining  constant,  and  the  hole  being  gradually 
rings.  brought  nearer  to  the  former. 


731. 

Fresnel's 
analysis  of 
this  case. 


24.00 
18.00 

13.50 

10.00 
9.25 
9.10 
8.75 
8.36 
8.00 
7.75 
7.00 
6.63 

6.00 

5.85 
5.50 
5.00 
4.75 
4.50 
4.00 
3.85 
3.50 


Central  Spot. 


White 
White 

Yellow 

Very  intense  orange 
Deep  orange  red 
Brilliant  blood  red 
Deep  crimson  red 
Deep  purple 
Very  sombre  violet 
Intense  indigo  blue 
Pure  deep  blue 
Sky  blue 

Bluish  white 

Very  pale  blue 

Greenish  white 

Yellow 

Orange  yellow 

Scarlet 

Red 

Blue 

Dark  blue 


Surrounded  by 


Rings  as  in  the  foregoing  Article. 

The  two  first  rings  confused,  the  red  of  the  3rd  and  green  of  the  4th 

orders  splendid. 
Interior  rings  much  diluted,  the  4th  and  5th  greens  and  3rd,  4th  and  5th 

reds  the  purest  colours. 
All  the  rings  are  now  much  diluted. 
The  rings  all  very  dilute. 
The  rings  all  very  dilute. 
The  rings  all  very  dilute. 
The  rings  all  very  dilute. 
A  broad  yellow  ring. 
A  pale  yellow  ring. 
A  rich  yellow. 

A  ring  of  orange,  from  which  it  is  separated  by  a  narrow,  sombre  space. 
r  Orange  red,  then  a  broad  space  of  pale  yellow,  after  which  the  other  rings 
\          are  scarcely  visible. 
A  crimson  red  ring. 

Purple,  beyond  which  yellow  verging  to  orange. 
Blue,  orange. 

Bright  blue,  orange  red,  pale  yellow,  white. 
Pale  yellow,  violet,  pale  yellow,  white. 
White,  indigo,  dull  orange,  white. 
White,  yellow,  blue,  dull  red. 
Orange,  light  blue,  violet,  dull  orange. 


The  series  of  tints  exhibited  by  the  central  spot  is,  evidently,  so  far  as  it  goes,  that  of  the  reflected  rings  in  the 
colours  of  thin  plates.  The  surrounding  colours  are  very  capricious,  and  appear  subject  to  no  law.  They  depend, 
indeed,  on  very  complicated  and  unmanageable  analytical  expressions,  with  which  we  shall  not  trouble  the  reader, 
but  content  ourselves  with  presenting  the  explanation  given  by  M.  Fresnel  of  the  changes  of  tint  of  the  central  spot 
in  white  light,  and  its  alternations  of  light  and  total  darkness  observed  by  him  in  an  homogeneous  illumination. 
Let  then  a  and  b  be  the  distances  of  a  small  hole  whose  radius  is  r  from  the  luminous  point,  and  a  screen 
placed  behind  the  hole  perpendicularly  to  the  ray  passing  directly  through  its  centre.  Then  if  we  consider  any 
infinitely  narrow  annulus  of  the  hole  whose  radius  is  2,  and  breadth  d  z,  this  annulus  will  send  to  the  central 
point  of  the  screen  a  system  of  waves  whose  intensity  is  proportional  to  the  area  of  the  annulus,  or  2  it  z  dz, 
but  whose  phase  of  undulation  differs  from  that  of  the  central  ray,  by  reason  of  the  difference  of  the  paths 
described  by  them.  Now,  calling/"  the  distance  of  each  point  in  the  annulus  from  the  centre  of  the  screen,  we 
have  f'  =  6*  -f-  z*,  and,  in  like  manner,  if  ff  be  the  distance  of  the  luminous  point  from  the  same  annulus, 
f3  =  a4  +  x*,  so  that  (/  +  /')  —  (a  +  6)  the  difference  of  paths,  or  interval  of  retardation,  is  equal  to 

— -( —  H )  =  — : — J— -.     Hence,  the  general  expression  in  Art.  632  for  the  amplitude  of  the  total  wave, 

2\ffl          6  /  '2  a  b 

incident  on  the  centre  of  the  screen  in  this  particular  case,  is  equivalent  to 

/-,  M      «*(«  +  *)!. 

*-J'  IT          2  a  6  X     j 


LIGHT.  485 

r,  integrating,  which  from  the  peculiar  form  of  the  differential  is  in  this  case  easy,  P»«  HI. 

a  b  \    C  (  t         z*  (a  +  6)\  ) 

x  =         |const  +  cos  *  '  •  (T  ~  -tt 


which,  extended  from  z  =  0  to  z  =  r,  gives 

_  ab\      (      .  ,  /  t        (a+b. 

\     __ 


a-)-  4 
a  b  \    C  .    •*  (a  +  6)  r*  t        /       *•(«+  6) 


6 


<     ,    / 

srT  +  (c 


This  expresses,  as  we  have  before  remarked  in  a  similar  case,  (Art.  718,)  two  partial  waves  differing  by  a  quarter- 
undulation,  and  expressing  it,  as  in  that  case,  by  X  =  X'  .  cos  0  -f-  X"  .  sin  0,  where  0  =  —  ,  we  find  for  the 
intensity  A*  of  their  resultant 

A.  =  X*  +  X*  =  4 

\a  +  b 

To  make  use  of  this,  however,  we  must  compare  it  with  what  would  be  the  direct  illumination  of  the  centre       732- 
of  the  screen,  if  the  aperture  were  infinite,  i.  e.  if  the  direct  light  from  the  luminous  point  shone  full  upon  it.  gj^^j^ 
To  this  case,  however,  neither  our  formula  nor  our  reasoning  are  applicable  ;   for  if  we  make  r  infinite  in  this  centra[  sl,ot 
expression,  it  becomes  illusory,  and  presents  no  satisfactory  sense,  and  in  our  reasoning  we  have  neglected  to  compared 
consider  the  law  of  diminution  of  the  intensity  of  the  oblique  waves,  or  regarded  0  (0)  in  Art.  631  as  invariable,  with  the 
which  in  this  extreme  case  is  far  from  the  truth.     We  must,  therefore,  have  recourse  to  another  method.     Now,  ^J^11""1'" 
M.  Fresnel  has  demonstrated   (and  our  limits  oblige  us  to  take  his  demonstration  for  granted)  that   this  total  Fresne|.s 
illumination  is  equal  to  one-fourth  of  that  which  the  centre  of  the  screen  would   receive  from  an   opening  of  theorem. 
such  a  radius,  that  the  difference  of  routes  of  a  ray  passing  through  the  centre,  and  one  diffracted  at  the  circum- 

ference, shall  be  an  exact  semi-undulation,  i.  e.    in  which  --  =  -  =  —  ,  or  r  =  V/  -    ;  —  j.     If  then  we 

•2a  b  2  v     a  -j-  o 

substitute  this  for  r  in  the  above  formula,  and  put  C  for  the  whole  illumination,  we  get,  on  the  same  scale, 

_       /tf&XV    /.     TrV 
C  =  I  -   I  .  (  sin  —    I   =  . 
\a  +  b)    \       -2  ) 

and,  consequently, 


2  ab 


/  /3\2 

In  this  expression  r,  a,  b  are  independent  of  X,  and  therefore  the  value  of  As  is  of  the  form  4  C  (  sin  2  ir  .  -  }       73J 

(a  -1.  M  rs  those  of  the 

where  B  =  -  —  -  —  -  —  .     Hence,  if  we  suppose  ligrht  of  all  colours  to  emanate  from  the  luminous  point,  the  reflected 
4  a  6  rin»s. 

r         /  p  Yl 

compound  tint  produced  in  the  central  point  of  the  screen  will  be  represented  by  S  -|  4  C  .  f  sin  2  <7r  —  )  r  and 

fa  _j_  ft)  ft 


-  — 


will  therefore,  by  Art.  673,  be  the  same  with  that  reflected  by  a  plate  of  air  whose  thickness  is  B,  or 

which  increases  as  6  diminishes  when  a  -f-  b  remains  constant.  Thus  we  see  the  origin  of  the  succession  of 
colours  of  the  central  spot  in  the  Table  above  recorded,  which  is  the  more  satisfactory,  as  that  experiment  was 
made  without  reference  to,  and  indeed  in  ignorance  of,  this  elegant  application  of  M.  Fresnel's  general  principles, 
the  merit  of  which  is  due  (as  he  himself  states)  to  M.  Poisson.* 

Another  very  curious  result  of  M.  Poisson's  researches  is  this,  that  the  centre  of  the  shadow  of  a  very  small       734. 
circular  opaque  disc,  exposed  to  light  diverging  from  a  single  point,  is  precisely  as  much   illuminated  by  the  Poisson's 
diffracted  waves  as  it  would  be  by  the  direct  light,  if  the  disc  were  altogether  removed.     We  cannot  spare  room  |!leo.r,?m  '"ol 
for  the  demonstration  of  this  singular  theorem.     It  has  been  put  to  the  test  of  experiment  by  M.  Arago,  with  Mtioa  i™th« 
a  small  metallic  disc  cemented  on  a  very  clear  and  homogeneous  plate  of  glass,  and  with  full  success.  centre  of  -<-. 

When  the  light  is  transmitted  through  two  equal  apertures,  placed  very  near  each  other,  the  rings  are  formed  small  cirrn- 
about  each  as  in  the  case  of  one  ;  but  besides  these  arise  a  set  of  narrower,  straight,  parallel  fringes  bisecting  l»r»h«dnw. 
the  interval  between  their  centres,  and  at  right  angles  to  the  line  joining  them.     If  the  apertures  be  unequal,       •  ^^>- 
these  fringes  assume  the  form  of  hyperbolas,    having    the  aperture  in  their  common  focus.      Besides  these  f^0°n 
also  two  other  sets  of    parallel  rectilinear  fringes  (in  the  case  of    equal  apertures)  go    off  in    the  form    of  throu«!i  two 
a  St.  Andrew's  cross  from  the  centre  at  equal  angles  with  the  first  set.     See  figures  147,  148.     When  the  apertures 
apertures  are  more  numerous  or  varied  in  shape,  the  variety  and  beauty  of  the  phenomena  are  extraordinary  ;  v"y  near 
but  of  this  more  presently.  ^a.ch  °^er- 

M.  Fresnel  has  shown,  that  when  the  light  from  a  single  luminous  point  is  received   on  two  plane  mirrors  ai^'  Ii8 

*  The  coincidence  in  the  higher  orders  of  colours  wa«,  however,  in  our  experiments  less  complete,  and  especially  the  green  of  the  third 
order,  which  was  wanting  altogether  in  some  cases. 


486  LIGHT. 

Light,      very  slightly  inclined  to  each  other,  so  as  to  form  two  almost  contiguous  images,  if  these  be  viewed  with  a     Part  MI. 
*~~~^-'~~~'  lens,  there  will  be  seen  between  them  a  set  of  fringes  perpendicular  to  the  line   joining  them.     These    are  '*— • ~^j— J 

736.  evidently  analogous  to  those  produced  by  the  two  holes  in  the  experiments  last  described.     The  experiment  is 
experiment  ^e*'cate >  f°r  '^ tne  surfaces  of  the  reflectors  at  the  point  where  they  meet  be  ever  so   little,  the  one  raised 
with  two      above  or  depressed  below  the  other,  so  as  to  render  the  difference  of  routes  of  the  rays  greater  than  a  very  few 
mirrors         undulations,  no  fringes  will   be  seen.     But  it  is  valuable,  as  demonstrating  distinctly  that  the  borders  of  the 
inclined  to    apertures  in  the  preceding  experiment  have  nothing  to  do  with  the  production  of  the  fringes,  the  rays  being  in 
each  other,    this  case  abandoned  entirely  to  their  mutual  action  after  quitting  the  luminous  point.     An  exactly  similar  set  of 

fringes  is  formed  if,  instead  of  two  reflectors,  we  use  a  glass,  plane  on  one  side,  and  on  the  other  composed  of 
Fig.  149.     two  planes,  forming  a  very  obtuse  angle,  as  in  fig.  149.     This  being  interposed  between  the  eye-lens  E  and  the 
luminous  point  S,  forms  two   images  S  and  S'  of  it ;   and  the  interference  of  the  rays  S  E  and  S'  E  from  these 
images,  forms  the  fringes  in  question. 

737.  Since  the  production  of  the  fringes  and  their  places  with  respect  to  the  images  of  the  luminous  point,  depends 
Effect  of      on  the  difference  of  routes  of  the  interfering  rays,  it  is  evident,  that  if,  without  altering  their  paths,   we  alter 
a^nser""   '^e  ve^oc^y  °f  one  °f  them  with  respect  to  the  other,  during  the  whole  or  a  part  of  its  course,  we  shall  produce 
medium  in    l'le  sarne  effect.     Now,  the  velocity  of  a  ray  may  be  changed  by  changing  the  medium  in  which  it  moves.     In  the 
one  of  two   undulatory  system,  the  velocity  of  a  ray  in  a  rarer  medium  is  greater  than  in  a  denser.    Hence,  if  in  the  path  of  one 
interfering    of  two  interfering  rays  we  interpose  a  parallel  plate  of  a  transparent  medium   denser  than  air,  (at  right  angles 

to  the  ray's  course,)  we  shall  increase  its  interval  of  retardation,  or  produce  the  same  effect  as  if  its  course  had 
been  prolonged.     If  then  a  thick  plate  of  a  dense  medium,  such   as  glass,  be  interposed   in   one  of  the  rays 
which  form  visible  fringes,  they  will  disappear  ;  because  the  interval  of  retardation  will  be  thus  rendered  suddenly 
equal  to  a  great  number  of  undulations,  whereas  the  production  of  the  fringes  requires  that  the   difference  oi 
routes  shall  be  very  small.     If,  however,  only  a  very  thin  lamina  be  interposed,  they  will   remain  visible,  but 
Fig.  150.     shift  their  places.     Thus,  in  fig.  150,  let  S  A,  S  B  be  rays  transmitted  through  the  small  apertures  A,  B  from  the 
luminous  point  S,  and  received  on  the  screen  D  C  E,  these  forming  a  set  of  fringes  of  which  C,  the  middle  one, 
Displace-     will  be  white.    Let  D,  E  be  the  dark  fringes  immediately  adjacent  on  either  side ;  and  things  being  thus  disposed, 
inert  of  the  let  a  thin  film  of  glass  or  mica  G  be  interposed  in  one  of  the  rays  S  A,  its  thickness  being  such  that  the  ray  in 
cx^neJ  ''  *ravers'n£  i'  shall  just  be  retarded  half  an  undulation.     Then  will  the  rays  A  E,  B  E,  which  before  were  in  com- 
plete discordance,  be  now  in  exact  accordance,  and  there  will  be  formed  at  E  a  bright  fringe  instead  of  a  dark 
one.     On  the   other  hand,  the  ray  AC  will   now  be  half  an   undulation  behind   BC,  instead    of  in   complete 
accordance  with  it,  so  that  at  C  there  will  be  formed  a  dark  fringe,  and  so  on.     In  other  words,  the  whole 
system  of  fringes  will  be  formed  as  before,  but  will  have  shifted  its  place,  so  as  to  have  its  middle  in  E  instead 
of  in  C,  i.  e.  will  have  moved  from  the  side  on  which  the  plate  of  the  dense  medium  is  interposed.     It  is  evident, 
that  if  the  plate  G  be  thicker,  the  same  effect  will  take  place  in  a  greater  degree. 

738.  To  make  the  experiment,  however,  it  must  be  considered  that  the  refractive  power  of  glass,  or  indeed  of  any  .-• 
Mode  of        but  gaseous  media,  is  so  great,  that  any  plate  of  manageable  thickness  would  suffice  to  displace  the  fringes  so 
to'th'e'le'st'of  ^ar  as  to  tnrow  them  wholly  out  of  sight.     But  we  shall  succeed,  if,  instead  of  a  single  plate  G  placed  over  one 
experiment,  aperture  A,  we  place  two  plates  G,  g  of  very  nearly  equal  thicknesses,  (such  as  will  arise  from  two  nearly  con- 
tiguous fragments  of  one  and  the  same  polished  plate,)  one  over  each  aperture  ;  or  we  may  vary  the   thickness 
of  the  plate  traversed  by  either  ray  by  inclining  it,  so  as  to  bring  it  within  the  requisite  limits.     This  done,  the 
effect  observed  is  precisely  that  described  ;  the  fringes  shift  their  places  from  the  thicker  plate,  without  sustaining 

Anument     any  alteration  in  other  respects.     This  elegant  experiment  affords  a  strong  indirect   argument  in  favour  of  the 
against  the    undulatory  system,  and  in  opposition  to  that  of  emission,  since  it  proves  that  the  rays  of  light  are  retarded  in 
lr  their  passage  through  denser  media,  agreeably  to  what  the  undulatory  system  requires,   and  contrary  to   the 
conclusions  of  the  corpuscular  doctrine. 

739.  MM.  Arago  and  Fresnel  have  taken  advantage  of  this  property,  to  measure  the  relative  refractive  powers  of 
Ara<?o  and    different  gases,  or  of  the  same  in  different  states  of  temperature,  pressure,  humidity,  &c.     It  is  manifest,  that  if 
method'of     any  considerable  portion  of  the  path  of  one  of  the  interfering  rays  be   made  to  pass  through  a  tube  closed  at    j 
detUminino-  both-ends  with  glass  plates,  and  the  other  through  equal  glass  plates  only,  the  fringes  will  be  formed  as  usual, 
refractions'  But  if  the  tube  be  exhausted,  or  warmed,  or  cooled,  or  filled  with  a  gas  of  different  refractive  density,  a  displace- 

•;f  ga»es.  ment  of  the  fringes  will  take  place,  which  (if  they  be  received  in  the  focus  of  a  micrometer)  may  be  measured 
with  the  greatest  delicacy.  Knowing  the  amount  of  their  displacement,  as  compared  with  the  breadth  of  the 
fringes,  we  know  the  number  of  undulations  gained  or  lost  by  one  ray  on  the  other ;  and  hence,  knowing  the 
internal  length  of  the  tube,  we  have  the  ratio  of  the  refracting  power  of  the  medium  it  contains  to  that  of  air. 
What  renders  this  method  remarkable  is,  that  there  is  actually  no  conceivable  limit  to  the  precision  of  which  it  in 
susceptible,  since  tubes  of  any  length  may  be  employed,  and  micrometers  of  any  delicacy. 

740.  The  phenomena  of  diffraction,  and  those  arising  from  the  mutual  interference  of  several  very  minute  pencils 
Fraunhofer's  of  rays  emanating  from  a  common  origin,  have  been  investigated  by  M.  Fraunhofer  with  great  care  and  extra- 
experiments  ordinary  precision,  by  the  aid  of  a  very  delicate  apparatus  devised  and  executed  by  himself. 

This   apparatus  consisted  of  a  repeating,    12-inch  theodolite,  ,-eading  to  every  4",  carrying,  attached  to  its 

terferelice""  horizontal  circle,  a  plane  circular  disc  of  six  inches  in  diameter,  having  its  axis  precisely  coincident  with  that  of 

His  appa-     the  theodolite,  and  having  its  own  particular  divisions  independent  of  those  of  the  theodolite.     In  the  centre  of 

ntus.  this  disc  was  placed  vertically  a  metallic  screen,  having  in  it  one  or  more  narrow,  vertical,  rectangular  slits,  or 

other  apertures,  and  so  fixed  as  to  have  the  middle  of  its  aperture,  or  system  of  apertures,  exactly  coincident  with 

the  axis  of  the  instrument.     Attached  to  the  great  circle  of  the  theodolite,  horizontally,  was  a  telescope,  having  its 

object-glass  three  inches  and  a  half  from  the  centre,  and  its  axis  directed  exactly  to  it,  and  precisely  parallel  to 

the  plane  of  the  limb,  and  provided  with  a  delicate  micrometer,  whose  parallel  threads  were  exactly  vertical. 


LIGHT.  487 

Light.      The  instrument  being  insulated   on  a  support  of  stone,  a  beam  of  solar  light  was  directed    by  a    heliostat,     Part  III. 
~-^s~~s  through  a  very  narrow  slit,  also  exactly  vertical,  having  a  breadth  of  one  hundredth  of  an  inch,  and  distant  463J  v>— •v^»/ 
inches  from  the  centre  of  the  theodolite,  so  as  to  fall  on  the  screen,  and,  being  transmitted  through  its  apertures, 
to  be  received  into  the  telescope.     It  is  manifest  that  the  eye-glass  of  the  telescope  will  here  view  the  fringes,  &c. 
as  they  are  formed  in  its  focus.     The  magnifying  power  of  the  telescope  used  by  Fraunhofer  varied  from  30  to 
50  times. 

M.  Fraunhofer  first  examined  the  effect  produced  by  the  diffraction  of  the  light  through  a  single  slit, — the       741. 
breadth  of  which  he  first  determined  with  the  greatest  precision  by  means  of  a  micrometer-microscope,  with  FrinSes 
which  he  assures  us  that  he  found  it  practicable  to  appreciate  so  minute  a  quantity  as  1 -50,000th  of  an  inch.  The  a™*1  °ig     y 
slit  being  then  placed  on  the  apparatus,  and  accurately  adjusted  before  the  object-glass  of  the  telescope,  which  narrow 
was  directed  exactly  to  the  aperture  in  the  heliostat,  the  image  of  the  latter  was  formed  in  its  focus,  accompanied  aperture. 
by  lateral  fringes,  which  by  the  effect  of  the  magnifying  power  were  dilated  into  broad  and  brilliant  prismatic 
spectra.     The   distances  of  the  red  ends  of  these  spectra  from  the   middle  point,  or  white  central  image,  were 
then  measured  accurately  by  means  of  the  micrometer.     The   result  of  a  great  number  of  experiments  with 
apertures  of  all  breadths  from  one-tenth  to  one-thousandth  of  an  inch,  agreed  to  astonishing  precision  with  each 
other,  and  with  the  following  laws,  viz.  that  (under  the  circumstances  of  the  experiment,) 

1.  The  angles  of  deviation  of  the  diffracted  rays,  forming  similar  points  of  the  systems  of  fringes  produced  Their  laws 
by  different  apertures,  are  inversely  as  the  breadths  of  the  apertures. 

2.  That  the  distances  of  similar  rays  (the  extreme  red,  for  instance,)  from  the  middle  in  the  several  spectra,  s" 
constituting  the  successive  fringes,  form  in  each  case  an  arithmetical  progression  whose  difference  is  equal  to  its 

Jirst  term. 

3.  That  calling  7  the  breadth  of  the  aperture,  in  fractions  of  a  Paris  inch,  the  angular  distances  L',  L",  L'", 
&c.  in  parts  of  a  circular  arc  to  radius  unity,  of  the  extreme  red  rays  in  each  fringe  from  the  middle  line,  are 

respectively  represented  by  L'  =  — ,  L"  =  2  .  — ,  L'"=3  .  — ,  &c.  where  L  =  0.0000211,  and  a  similar  law 

7  7  7 

holds  for  all  the  other  coloured  rays,  different  values  being  assigned  to  L  for  each. 

This  conclusion  agrees  perfectly  with  the  result  of  an  experiment  related  by  Newton  in  the  Hid  Book  of  his        74'^. 
Optics.  He  ground  two  knife  edges  truly  straight,  and  placed  them  opposite  to  each  other,  so  as  to  be  in  contact  Newton's 
at  one  end,  and  at  the  other  to  be  at  a  small  distance,  such  that  the  angle  included  between  them  was  about  e*P"" 
1°  54',  thus  forming  a  slit  whose  breadth  at  their  intersection  was  evanescent,  and  at  4  inches  from  that  point  ^n-l!e  e^es 
|th  of  an  inch,  and  in  the  intermediate  points,  of  course,  of  every  intermediate  magnitude.     Exposing  this  in  a 
sunbeam   emanating  from  a  very  small   hole  at  15  feet  distance,  he  received  their  shadows   on  a  white  screen 
behind  them,  and  observed  that  when  they  were  received  very  near  to  the  knife  edges,  (as  at  half  an  inch,)  the 
fringes  exterior  to  the  shadow  of  each  edge  ran  parallel  to  its  border  without  sensible  dilatation,  till  they  met  and 
joined  without  crossing,  at  angles  equal  to  that  contained  between  the  knife  edges.     But  when  the  shadows  were 
received  at  a  great  distance  from  the  knives,  the  fringes  had  the  form  of  hyperbolas,  having  for  one  asymptote  the 
shadow  of  the  knife  to  which  they  respectively  belonged,  and  for  the  other  a  line  perpendicular  to  that  bisecting  the 
angle  of  the  two  shadows,  each  fringe  becoming  broader  and  more  distinct  from  the  shadow  which  it  bordered,  as  it 
approached  the  angle.     These  hyperbolas  crossed  without  interfering,  as   represented  in  fig.  151.     Their  points  F'£- IS1 
of  crossing,  Newton  found,  however,  not  to  be  at  a  constant  distance  from  the  angle  included  between  the  pro- 
jections  of  the  knife  edges,  but  to  vary  in  position  with  the  distance  from  the  knives,  at  which  the  shadow  is 
received  on  the  screen  ;   and  hence,  he  says,  "  I  gather  that  the  light  which  makes  the  fringes  upon  the  paper,  is 
not  the  same  light  at  all  distances  of  the  paper  from  the  knives  ;  but  when  the  paper  is  held  very  near  the  knives, 
the  fringes  are  made  by  light  which  passes  by  their  edges  at  a  less  distance,  and   is  more  bent  than  when  the 
paper  is  held  at  a  greater  distance  from  the   knives."     Newton,  however,  left   these  curious   researches,  which 
could  hardly  have  failed  to  have  led  in  his  hands  to  a  complete  knowledge  of  the  principles  of  diffraction — unfinished ; 
being,  as  he  says,  interrupted  in,  and  unwilling  to  resume  them  :  doubtless,  owing  to  the  chagrin  and  opposition 
his  optical  discoveries  produced  to  him.     An  unmeet  reward,  it  must  be  allowed,  for  so  noble  a  work,  but  one  of 
which,  unhappily,  the  history  of  Science  affords  but  too  many  parallels. 

The  above  were  the  results  obtained  by  M.  Fraunhofer  when  the  two   edges  of  the  aperture  were  both  in  a       743. 
plane  perpendicular  to  the  incident  rays  ;  but  when  the  same  effective  breadth  was  procured,  by  inclining  a  larger  Case  wtlcn 
aperture  obliquely,  so  as  to  reduce  its  actual  breadth  in  the  ratio  of  the  cosine  of  its  incidence  to  radius,  or  by  '!' 
limiting  the  incident  ray  by  two  opaque  edges  at  different  distances  from  the  object-glass  of  the  telescope,  the  We,cP-f 
phenomena  were  very  different.     To  accomplish  this,  two  metallic  plates  were  fixed  upright  on  the  horizontal  different 
plate  of  the  theodolite,  having  their  edges  exactly  vertical,  and  precisely  at  opposite  extremities  of  a  diameter,  distances 
Then,  by  turning  the  plate  round  on  its  axis,  the  passage  allowed  to  the  light  between  them  could  be  increased  or  fro™  the. 
diminished  at  pleasure.     The  phenomena,  then,  were  as  follows.     When  the  opening  allowed  to  the  light  was  u,r'?j"  £,' 
considerable,  as  0.02  or  0.04  inch  (Paris,)  the  fringes  were  exactly  similar  to  those  observed  when  the  edges  were 
equidistant  from  the  object-glass  ;  but  as  the  opening  diminished,  they  ceased  to  be  symmetrical  on  both  sides  of 
the  middle  line,  those  on  the  side  of  that  edge  of  the  aperture  nearest  to  the  telescope  becoming  broader  than 
those  on  the  other,  which,   on  their  part,  undergo    no  sensible  alteration.       As   the  aperture  contracts,  this 
inequality  increases,  till   at  length  the  dilated  fringes  begin  to  disappear  in  succession,   the   outermost  first, 
which  they  do  by  suddenly  acquiring  an  extraordinary  magnitude,  so  as  to  fill  the  whole  field  of  the  telescope, 
and  thus,  as  it  were,  losing  themselves.     While  these  are  thus  vanishing,  those  on   the  other  side  remain  quite 
unaltered  till  the  last  is  gone,  when  they  all  disappear  at  once,  which  happens  at  the  moment  that  the  opening 
is  reduced  to  nothing  by  the  two  edges  covering  each  other. 


488  LIGHT. 

Light.         When  the  aperture  placed  before  the  object-glass,  instead  of  being  a  straight  line,  was  a  small,  circular  hole, 
>«— •v^**'  and  the  aperture  of  the  heliostat,  in  like  manner,  a  minute  circle,  the  phenomena  of  the  rings  were  observed,  and 

744.  their  diameters  could  be  accurately  measured  by   the   micrometer.     The   results  of  these   measurements  led 
Case  of  a     M.  Fraunhoftr  to  the  following  laws :   1st,  that  for  apertures  of  different  diameters,  the  diameters  of  the  rings 
small.circu-  are  inversely  as  those  of  the  apertures  forming  them  ;    2dly,  that  the  distances  from  the  centre  of  the  maxima 

re'  of  extreme  red  rays  (or  of  rays  of  any  given  refrangibility)  in  the  several  rings  of  one  and  the  same  system,  form 
an  arithmetical  progression,  whose  difference  is  somewhat  less  than  its  first  term.     Thus,  calling  7  the  diameter 

of  the  aperture,  and  putting  L  =  — —  and  I  ==  — ,  he  found  L'  =  I,  L"  =  I  +  L,  L'"  =  I  -f-  2  L, 

7  7 

&c.,  where  L'  L",  &c.  represent  the  angular  semidiameters  of  the  several  rings  expressed  in  arc  of  a  circle  to 
radius  unity.  The  near  coincidence  of  the  value  of  L  in  this  case,  with  that  in  the  case  of  a  linear  aperture,  and 
the  small,  but  decided  difference  of  the  values  of  the  first  term  of  the  progression  in  the  two  cases,  are  very 
remarkable. 

745.  When  the  aperture  was  a  very  narrow,  circular  annulus,  such  as  might  be  traced  with  a  steel  point  on  a  gilt 
Case  of  a     disc  of  glass,  of  whatever  diameter,  the  image  was  a  circular  spot,  surrounded  in  like  manner  by  coloured  rings, 
very  narrow,  the  diameters  of  which  depended  nowise  on  the  diameter,  but  only  on  the  breadth  of  the  annulus,  being  in  fact 

(as  might  be  expected)  the  very  same  as  the  intervals  between  similar  opposite  fringes,  on  both  sides  of  the 
central  line  in  the  image  produced  by  a  linear  aperture  of  equal  breadth. 

746.  But  the  most  curious  parts  of  M.  Fraunhofer's  investigations  are  those  which  relate  to  the  interference  of  rays 
Interference  transmitted   through  a  great  many  narrow  apertures  at  once.     When  these  apertures  are  exactly  equal,  and 
of  many  rays  p]aced  at  exactly  equal  distances  from  one  another,  phenomena  of  a  totally  different  kind  from  those  originating 

in  a  single  aperture  are  seen.  In  his  first  experiments  of  this  kind  he  formed  a  grating  of  wire,  by  stretching 
gratings.  a  veT  fine  w're  across  a  frame,  in  the  form  of  a  narrow,  rectangular  parallelogram,  whose  shorter  sides  were 
screws  tapped  in  the  same  die,  and  therefore  precisely  similar ;  across  these  screws  in  the  consecutive  intervals 
between  their  threads  the  wires  were  stretched,  and  of  course  could  not  be  otherwise  than  parallel  and  equidistant. 
The  diameter  of  the  wire  was  0.002021  Paris  inch,  the  intervals  between  them  each  0.003862,  and  the  grating 
consisted  of  260  such  wires.  When' this  apparatus  was  placed  precisely  vertical  before  the  object-glass  of  his 
telescope,  and  illuminated  by  a  narrow  line  of  light  0.01  inch  in  breadth,  also  exactly  vertical,  forming  the  aper- 
ture of  the  heliostat,  the  image  of  this  was  seen  in  the  telescope,  colourless,  well  defined,  and  in  all  respects  pre- 
cisely as  it  would  have  been  seen  without  the  interposition  of  any  grate  or  aperture  at  all,  occupying  the  centre  of 
Spectra  of  the  field,  only  less  bright.  On  either  -side  of  this  was  a  space  perfectly  dark,  after  which  succeeded  a  series  of 
the  second  prismatic  spectra,  which  he  calls  spectra  of  the  second  class,  not  consisting  of  tints  melting  into  each  other, 
according  to  the  law  of  the  coloured  rings,  or  any  similar  succession  of  hues  depending  on  a  regular  degra- 
dation of  light,  but  of  perfectly  homogeneous  colours  ;  so  much  so,  as  to  exhibit  the  same  dark  lines  crossing  them 
as  exist  in  the  purest  and  best  defined  prismatic  spectrum.  In  the  disposition  of  things  already  described,  the 
first,  or  nearer  spectrum  is  completely  insulated,  the  space  between  it  and  the  central  image,  as  well  as 
between  it  and  the  second  spectrum,  being  quite  dark.  The  violet  ends  of  the  spectra  are  inwards,  and  the  red 
outwards  ;  but  the  violet  end  of  the  third  spectrum  is  superposed  on  the  red  end  of  the  second,  so  as  in  place 
of  a  dark  interval  to  produce  a  purple  space ;  and  as  we  proceed  farther  from  the  middle,  the  spectra  become 
more  and  more  confounded,  but  not  less  than  thirteen  may  easily  be  counted  on  each  side  by  the  aid  of  a  prism 
refracting  them  transversely,  so  as  to  separate  their  overlapping  portions. 

747.  The  measurement  of  the  distances  of  similar  points  in  the  several  spectra  are  rendered   susceptible  of  the 
Ratio  of  the  utmost  precision  by  means  of  the  dark  lines  which  cross  them.     A  very  remarkable  peculiarity  of  these  spectra 

must,  however,  be  here  noticed,  viz.  that  although  the  dark  lines  hold  exactly  the  same  places  in  the  order  of 
colours,  or,  in  other  words,  correspond  to  precisely  the  same  degrees  of  refrangibilily,  as  in  the  prismatic  spectra 
formed  by  refraction,  yet  the  ratio  of  the  intervals  between  them,  or  the  breadths  of  the  several  coloured  spaces, 
differ  entirely  in  the  two  cases.  Thus,  in  the  diffracted  spectra,  the  interval  between  the  lines  C  and  D  (fig.  94) 
is  very  nearly  double  of  that  between  G  and  H,  while  in  a  spectrum  formed  by  a  flint-glass  prism  of  an  angle 
of  270,  the  proportion  is  reversed,  and  in  a  water  prism  of  the  same  angle  CD  :  G  H  :  :  2  :  3. 

748.  In  the  diffracted  fringes  formed  by  a  single  aperture,  their  distances  (as  we  have  seen)  from  the  axis  depends 
Their  laws.  OI1]y  on  tne  breadth  of  the  aperture,  being  inversely  as  that  breadth.     In  the  spectra  formed  by  a  great  "number, 

their  distances  from  the  central  image  depends  neither  on  the  breadths  of  the  apertures  nor  on  the  intervals 
between  them,  but  on  the  sum  of  these  quantities,  that  is,  on  the  distances  between  the  middle  points  of  the 
consecutive  apertures,  (or,  in  the  case  before  us,  on  the  distances  between  the  axes  of  the  wires.)  By  a  series  of 
measures  performed  with  the  utmost  care  and  precision  on  wire  gratings  of  a  great  variety  of  dimensions, 
M.  Fraunhofer  ascertained  the  following  laws  and  numerical  values. 

749.  1.  For  different  gratings,  if  we  call  7  the  breadth  of  each  of  the  interstices  through  which  the  light  passes,  and 
£  that  of  each  of  the  opaque  intervals  between  them,  the  magnitudes  of  spectra  of  the  same  order,  and  the  dis- 
tances of  similar  points  in  them  from  the  axis,  is  inversely  as  the  sum  7-)-  e. 

750  2.  The  distances  of  similar  points,  (j.  e.  of  similar  colours  or  similar  fixed  lines,)  in  the   several  consecutive 
spectra  formed  by  one  and  the  same  grating  from  the  axis,  constitute  an  arithmetical  progression  whose  difference 
is  equal  to  its  first  term. 

751  3.  For  the  several  refrangibilities  corresponding  to  the  fixed  lines  B,  C,  D,  E,  &c.  the  first  term  of  this  pro- 
gression is  numerically  represented  by  the  respective  fractions  which  follow,  being  the  lengths  of  the  arcs,  or 
their  sines  to  radius  unity. 


LIGHT. 

Light  0.00002541  _  0.00001945  0.00001464 

—  NX—  "*  ^     -    -  i  -  "  -    '  -   -  i  -  $  -  '  - 

-— 


0.00002422  0.00001794 

C  =  -          -  ;         F  =      .  ;  &c. 

7  +  * 


7-f-o 


0.00002175  0.00001587 

D=-T+—  ;  T+r-; 

These  results  were  all,  however,  deduced  from  gratings  so  coarse  as  to  allow  of  our  regarding  the  angles  of      752. 
diffraction  as  proportional  to  their  sines;    but  when  extremely  fine  gratings    are    employed,    the    spectra  are  Case  of 
formed  at  great  distances  from  the  axis,  and  the  analogy  of  other  similar  cases,  as  well  as  theory,  would  lead  us  extremely 
to  substitute  sin  B,  sin  C,  sin  D,  &c.  in  the  place  of  B,  C,  D,  &c.     This,  M.  Fraunhofer  found  by   experiment  cg°^ln  s 
to  be  really  the  case.     The  construction  of  gratings  proper  for  these  delicate  purposes,  however,  was  no  easy 
matter.     Those  employed  by  him  were  nothing  more  than  a  system  of  parallel  and  equidistant  lines  ruled  on  Methods  of 
plates  of  glass  covered  with  gold-leaf,  or  with  the  thinnest  possible  film   of  grease  ;  by   the  former  of  these  constructing 
methods  he  found,  that  the  proximity  of  the  lines  might  be  carried  to  the  extent  of  placing  about  a  thousand  in  t"em- 
the  inch,  but  when  he  would  draw  them  still  closer,  the  whole  of  the  gold-leaf  was  scraped  off.     When  the  sur- 
face was  covered  with  a  film  of  grease  so  thin  as  to  be  almost  imperceptible  to  the  sight,  (although  the  intervals 
were  in  this  case  transparent,)  no  change  was  produced  in  the  optical  phenomena,  so  far  as  the  spectra  were 
concerned,  only  the  brightness  of  the  central  image  being  increased.     By  this  means  he  was  enabled  to  obtain 
a  system  of  parallel   lines  at  not  more  than  half  the  distance  from  each  other  that  could  be  produced  on  gold- 
leaf:  but  beyond  this  degree  of  proximity,  he  found  it  impossible  to  carry  the  ruling  of  equidistant  lines  on  any 
film  of  grease  or  varnish.     But  this  being  still  far  short  of  his  wishes,  he  had  recourse  to  actual  engraving  with 
a  diamond  point  on  the  surface  of  the  glass  itself,  and  by  this  means  was  enabled  to  rule  lines  so  fine  as  to  be 
absolutely  invisible  under  the  most  powerful  compound  microscope,  and  so  close  that  30,000  of  them  lie  in  a 
single  Paris  inch.     When  so  excessively  near,  however,  no  accuracy  of  machinery  will  ensure  that  perfect  equi- 
distance  which  is  essential  to  the  production  of  the  spectra  now  under  consideration,  and  he  found  it  impossible 
to  succeed  in  placing  them  nearer  than  0.0001223,  (or  about  8200  to  the  inch,)  with  such  a  degree  of  precision 
as  to  enable  him  to  distinguish  the  fixed  lines  in  the  spectra  ;   and,  if  it  be  considered,  that  a  deviation  to  the 
extent  of  the  hundredth  part  of  the  just  interval  frequently  occurring,  is  sufficient  to  obliterate  these,  and  that  to 
produce  the  spectra  in  sufficient  brightness  to  affect  the  eye,  some  hundreds  or  even   thousands   must  be  ruled, 
we  shall  be  enabled  to  form  some  conception  of  the  difficulties  to  be  encountered  in  researches  of  this   kind. 
For  a  detail  of  some  of  these,  and  of  the  methods  employed  by  him  to  count  their  number  and  measure  their  dis- 
tances, we  must  refer  to  his  original  Memoir,  (read  to  the  Royal  Bavarian  Academy  of  Sciences,  June  14,  1823.) 

In  the  course  of  these  researches,  M.  Fraunhofer  met  with  a  very  singular  and  instructive  peculiarity  in  one      753. 
of  the  engraved  glass-gratings  used  by  him;  which,  although  it  produced  spectra  equidistant  on  either  side  of  The  spectra 
the  axis,  yet   gave  always   those  on  one  side  a  much   greater  degree  of  brightness  than   those    on   the   other,  modified  by 
Attributing  this  to  the  form  of  the  furrows  being  sharper  terminated  on  one  side  than  on  the  other,  owing  either  '!' 
to  the  figure  of  the  diamond  point  or  the  manner  of  its  application,  he  endeavoured  to  produce  a  similar  struc-  the  gratings 
ture  of  the  striae  in  a  film  of  grease  spread  on  glass,  by  purposely  applying  the  engraving  tool  obliquely,  and  the 
attempt  proved  successful. 

When  the  incident  rays  from  the  opening  in  the  heliostat  fell  obliquely  on  the  grating,  it  might  be  supposed      754. 
that  the  phenomena  would  be  the  same  as  those  exhibited  by  a  closer  grating,  having  intervals  less  in  proportion  Case  of 
of  the  cosine  of  the  angle  of  incidence  to  radius.     But  the  analogy  of  the  unsymmetrical  fringes  produced  by  a  inclined 
single  aperture,  whose  sides  lie  in  a  plane  oblique  to  the  incident  ray,  may  lead  us  to  expect  a  different  result,  |yatm£s 
and  experiment  confirms  the  surmise;    thus,  M.  Fraunhofer  found,  that  on  inclining  a  grating,  whose   intervals  tn"^1" 
(7  -f-  £)  were  each  equal  to  0.0001223  inch,  so  as  to  make  the  angle  of  incidence  55°  with  the  perpendicular,  spectra  of 
the  distance  of  the  first  fixed  line  D  from  the  axis  on  the  one  side  of  the  axis  was  15°  6',  and  on  the  other  no  less  the  second 
than  30°  33',  or  more  than  double.  class- 

The  facts  deduced  by  M.  Fraunhofer  in  the  above  detailed  researches  are  certainly  extremely  curious.     The      755, 
most  interesting  and  remarkable  point  about  them  is  the  perfect  homogeneity  of  colour  in  the  spectra,  indicating  Theoretical 
a  saltus,  or  breach  of  continuity,  in  the  law  of  intensity  of  each  particular  coloured  ray  in  the  diffracted  beam,  considera- 
For  it  is  obvious,  that  taking  any  one  refrangibility  (that  corresponding  to  the  fixed  line  C,  for  example,)  the  tions- 
expression  of  its  intensity  in   functions  of  its  distance  from  the  axis  must  be  (analytically  speaking)  of  such  a 
nature  as  to  vanish  completely  for  every  value  of  that  distance,  excepting  for  a  certain  series  in  arithmetical  pro- 
gression, or,  as  it  is  called,  a  discontinuous  function  ;   so  that  the  curve  representing  such  value,  having  the 
distance  from  the  axis  for  its  abscissa,  must  be  a  series  of  points  arranged  above  the  axis  at  equal  intervals  ;  or, 
at  least,  a  curve  of  the  figure   represented  in  fig.  151,  in  which  certain  extremely  narrow  portions,  equidistantly 
arranged,  start  up  to  considerable  distances  from  the  axis,  while  all  the  intermediate  portions  lie  so  close  to  that 
line  as  to  be  confounded  with  it.     The  manner  in  which  such  a  function  can  be  supposed  to  originate  from  the 

summation  of  a  series  of  the  values  of  f  d  v  .  sin  —  v2  andy*  d  v  .  cos  -—  v',  (Art.  7  1  8,)  taken   successively  be- 

tween limits  corresponding  to  the  boundaries  of  the  several  interstices,  involves  too  many  complicated  conside- 
rations to  enter  into  in  this  place.     M.  Fraunhofer,  meanwhile,  states  the  following  general  expression,  as  the 
result  of  his  own   investigations   founded  on  the  principle   of  interferences.     Let  n  indicate  the  order  of  any 
VOL.  iv.  3  s 


490 


LIGHT. 


Fraun- 
liofer's 

formula. 


spectrum,  reckoned  from  the  axis  ;  e  the  distance  from  the  middle  of  one  interstice  to  that  of  the  adjacent  one 
=  "f  -f-  S  ;  X  the  length  of  an  undulation  of  an  homogeneous  ray  ;  a  the  angle  of  incidence  of  the  ray  from  the 
luminous  point  on  the  grating  ;  and  y  the  length  of  a  perpendicular  let  fall  from  the  micrometer  thread  of  the 
telescope,  (or  from  the  point  in  the  focus  of  its  object-glass,  where  that  particular  homogeneous  ray  in  that 
spectrum  is  found,)  on  the  plane  of  the  grating.  Then,  if  the  angular  elongation  of  that  ray  from  the  axis  be 
called  0("',  we  shall  have,  in  general, 


Pan  Hi. 


cotanP  '  = 


V  {  6*  -  (e  .  sin  a  +  n  X)a  } 
'  =  -  -  —  - 


{  4  y*  -f-  e*  -  (e  .  sin  a  -f  n  X)«  } 
-  —  -.  ;  -  —  -  —  - 

2  y  (e  .  sin  a  -j-  n  \) 


In  this  equation,  n  is  to  be  regarded  as  +  for  the  spectra  which  lie  on  the  side  of  the  axis  on  which  the  incident 
ray  makes  an  obtuse  angle  with  the  plane  of  the  grating,  and  negative  for  the  spectra  on  the  other  side.  This 
formula  he  states  to  be  rigorous,  and  independent  of  any  approximation.  When  y  is  very  great  (as  it,  in  fact, 
always  is,)  compared  with  e  and  X,  this  reduces  itself  simply  to 


.,„,        */e*  —  (e  .  Sin  a  -4-  M  V)8 
cotan  ffl">  =  -  _i  -  —£- 

e  .  sin  a  -f-  n  X 


or  sin     "  = 


e  .  sin  a  4-  n  X 


756. 

Lengths  of 
undulations 
of  the  rays 
B,C,D,&c. 
assigned  by 
Fraunhofer. 


757. 
Diffracted 
spectra  pro- 
duced by 
reflexion. 

758. 
Alleged 
limit  to  the 
powers 
»il  micro- 
scopes. 


759. 

Spectra 
produced 
by  compo- 
lite  gra- 
tings. 


Singular 
phenome- 
non noticed 
by  Fraun- 
hofer 
respecting 
the  inten- 
sity of  the 
spectra. 


760. 

Various 
stages  of 
the  pheno- 
mena. 
Spectra  of 
the  first  class 

761 


This  formula,  applied  to  M.  Fraunhofer's  measures  of  the  distances  of  the  same  fixed  lines  in  successive  spectra 
on  either  side  of  the  axis,  in  the  case  of  inclined  gratings,  represents  them  with  perfect  exactness.  When  the 

gratings  are  perpendicular  to  the  ray  a  =  0,  and  the  equation  becomes  sin  £(">  =  -  ,  which  is  the  law  before 

e 

noticed  for  symmetrical  spectra.  And  hence,  too,  it  appears  that  the  values  of  X,  or  the  lengths  of  the  undulations 
for  the  several  rays  designated  by  C,  D,  E,  &c.,  are  no  other  than  the  numerators  of  the  fractions  in  Art.  751, 
expressed  in  parts  of  a  Paris  inch,  which  thus  become  data  of  the  utmost  value  in  the  theory  of  light,  from 
the  great  care  and  precision  with  which  they  have  been  fixeds  and  for  the  possibility  of  identifying  them  at 
all  times. 

If  the  unruled  surface  of  the  glass  grating  be  covered  with  black  varnish,  and  the  light  reflected  from  the 
ruled  surface  be  received  in  the  telescope,  the  very  same  phenomena  are  seen  as  if  the  light  had  been  transmitted 
through  the  glass,  and  the  same  analytical  expression,  according  to  M.  Fraunhofer,  applies  to  both  cases. 

A  curious  consequence  of  this  expression  is,  that  if  e,  the  distance  between  the  lines,  be  less  than  \,  and  the 
light  fall  perpendicularly  on  the  grating,  so  that  sin  a  =  0,  we  shall  have  sin  0:"'  >  1,  and  therefore  <Xn)  imagi- 
nary. It  appears,  therefore,  that  lines  drawn  on  a  surface  distant  from  each  other  by  a  less  quantity  than  one 
undulation  of  a  ray  of  light,  produce  no  coloured  spectra.  Hence,  such  scratches,  or  inequalities,  on  polished 
surfaces,  have  no  effect  in  disturbing  the  regularity  of  reflexion  or  refraction,  and  produce  no  dimness  or 
mistiness  in  the  image  ;  if  less  distant  from  each  other  than  this  limit.  M.  Fraunhofer  seems  inclined  to 
conclude  further,  that  an  object  of  less  linear  magnitude  than  X  can  in  consequence  never  be  discerned  by 
microscopes,  as  consisting  of  parts  :  a  conclusion  which  would  put  a  natural  limit  to  the  magnifying  power  of 
microscopes,  but  which  we  cannot  regard  as  following  from  the  premises. 

When  the  intervals  of  the  parallel  interstices  are  unequal,  and  disposed  with  no  regularity,  the  light  of  the 
diffracted  spectra  of  different  combinations  is  confounded  together,  and  a  white  misty  streak  at  right  angles  to 
the  direction  of  the  lines  arises  ;  but  when  they  are  regularly  unequal,  so  that  the  same  intervals  recur  in 
regular  periods,  if  we  call  E  (=  <•'  -f-  c"  -f-  e'"  -f-  &c.)  the  interval  between  any  two  distant  by  a  whole  period, 

we  shall  have,  for  the  law  of  the  lateral  spectra,  the  equation  sin  &*>  =  -=-.     And  the  spectra  so  formed,  are 

15 

still  observed  to  consist  of  homogeneous  light,  exhibiting  the  fixed  lines  with  great  distinctness.  A  very  curious, 
and,  as  far  as  concerns  the  practical  measurement  of  the  phenomena,  useful  observation  has  been  made  by 
M.  Fraunhofer  on  the  spectra  so  formed  by  these  composite  gratings,  viz.  that  although  they  follow  the  same  law 
in  respect  of  their  distances  from  the  axis,  yet  the  successive  spectra  differ  greatly  in  intensity,  some  being  so 
faint  as  to  be  scarce  perceptible,  while  the  immediately  adjacent  ones  will  often  be  very  intense.  Owing  to 
this  cause,  spectra  of  the  higher  orders,  which  in  a  simple  grating  the  interval  of  whose  interstices  is  represented 
by  E,  are  confused  and  obliterated  by  the  encroachment  of  those  adjacent,  are  often  very  distinct  when  formed 
by  a  composite  grating,  the  period  of  recurrence  of  whose  similar  interstices  is  E  =  e'  -f-  e"  -f-  e'"  -f-  &c.  Thus, 
M.  Fraunhofer  was  never  able  through  a  simple  grating  to  see  the  fixed  lines  C  and  F  in  the  spectrum  of  the 
12th  order,  reckoning  from  the  axis,  while  in  a  composite  grating,  consisting  of  three  systems  of  lines  continually 
repeated,  whose  intervals  e',  e",  e'"  were  to  each  other  as  25  :  33  :  42,  these  fixed  lines  as  well  as  the  lines  D  and 
E,  were  distinctly  seen  in  the  12th  spectrum,  owing  to  the  almost  total  disappearance  of  the  10th  and  1  1th.  Nay, 
even  the  fixed  line  E  in  the  24th  spectrum  could  be  seen,  and  its  distance  from  the  axis  measured  with  this 
grating. 

Such  are  the  extreme  cases  of  the  phenomena  as  produced  by  a  single  aperture,  and  by  an  infinite,  or,  at 
least,  very  great  number  ;  but  the  intermediate  steps  and  gradations  by  which  one  set  of  phenomena  pass  into  the 
other,  remain  to  be  traced.  When  a  single  interstice  is  left  open  in  a  grating,  the  spectra  are  formed  as  described 
in  Art.  741.  These,  M.  Fraunhofer  calls  spectra  of  the  first  class,  and  their  colours  are  not  homogeneous,  but 
graduate  into  one  another. 

When  two  contiguous  interstices  are  left  open,  the  spectra  of  the  first  class  appear  as  before  ;  but  between  the 
axis  and  the  first  spectrum  on  either  side  appear  other  spectra,  which  M.  Fraunhofer  terms  imperfect  spectra  of 
the  second  class,  their  colours  being  similar  to  those  of  the  first  class,  and  no  fixed  lines  being  visible  in  them. 


LIGHT.  491 

Light       When  three  adjacent  interstices  are   left,  open,  a  third  set  of  spectra,  or  spectra  of  the  third  class,  are  formed    Tart  III. 
~"v*™</  between  the  axis  and  the  nearest  of  the  imperfect  spectra  of  the  second  class.     Besides  these,  no  new  classes  of  ^—^^^ 
spectra  arise  by  a  further  increase  of  the  number  of  interstices  ;   but  these  undergo  a  series  of  modifications  as  Spectra  of 
the  interstices  grow  more  numerous.     These  are  chiefly  as  follows: 

The  spectra  of  the  third  class  grow  narrower,  and  approach  the  axis,  till   at  last  they  run  together  and  form      ^gg 
by  their  union  the  colourless,  well-defined  image  of  the  opening  of  the  heliostat  in  the  axis  of  the  whole  pheno-  MoliifitTa- 
menon.     By  a  series  of  exact  measurements,  M.  Frannhofer  found  their  breadths  to  be  inversely  as  the  number  tionsof 
of  interstices  by  which  they  are  produced  in  the  same  grating,  and  inversely  as  the  intervals  of  the  interstices  for  these  spec- 
different  ones  ;  and  in  general,  that  7  -f-  c  =:  e  representing  this  interval,  m  the  number  of  interstices  used,  and  n  tra  ty  '"• 
the  order  of  the  spectrum,  0  n>  the  distance  of  extremity  of  the  red  rays  in  that  spectrum  is  given  by  the  equation  ^^bef  „(•' 

n  0.000020S  interfering 

0°°  =  —    X .  rays. 

Formula  fur 

As  the  spectra  of  the  third  class  contract  into  the  axis,  they  leave   a  dark   space  betwpen   it  and  the  first  spectra  of 
spectrum  of  the  second  class.    This  and  the  other  spectra  of  that  class  meanwhile  grow  continually  more  vivid  and  thlr^  C'a5s- 
homogeneous  in  respect  of  colour  ;   till  at  length,  when  the  number  of  interfering  rays  is  very  much  increased,  „ 
the  fixed  lines  begin  to  appear  in  them,  and  they  acquire  the  character  of  perfect  spectra  of  the  second  class.      from  jmper_ 
M.  Fraunhofer  next  examined  the  phenomena  produced  by  immersing  in  media  of  different  refractive  powers  feet  to  per- 
the  gratings  used,  when  he  found  all  the  phenomena  precisely  similar ;   but  the  distances  at  which  the  several  fec'  spectra 
spectra  were  formed  from  the  axis,  to  be  less  than  when  in  air,  in  the  inverse  ratio  of  the  refractive  indices. 

A  very  beautiful  and  splendid  class  of  optical  phenomena  has  been  investigated  and  described  by  M.  Fraun-  °  "^g* 
hofer,  which  arise  by  substituting  for  the  gratings  used  in  the  above  experiments  very  small  apertures  of  regular  phenomena 
figures,   such   as  circles  and  squares,  either  singly  or  arranged   in   regular  forms,  in  great  numbers;    as,  for  Of  gratings 
instance,  when  two  equal  wire  gratings  are  crossed  at  right  angles.     Fig.  151  is  a  representation  of  the  pheno-  immersed 
menon  produced  when  the  light  is  received  on  the  object-glass  of  the  telescope  through  two  circular  holes  of  the  '"  f"'1'8 
diameter  0.02227  inch,  placed   at  a  distance   of  0.03831   inch   centre  from  centre.     Each   compartment  is  a       ' 
separate  spectrum.     In  the  bands  a  a,  bb  we  see  here  plainly  the  origin  and  minute  structure  of  the  vertical  and  o|.u°slltut10" 
crossed  fringes   described  in  Art.  735.     The  appearances  vary  as  the  number  of  apertures  is   increased,  the  minute 
spectra  growing  purer  and  more  vivid.     That  which  arises  when  two  equal  wire  gratings  are  crossed,  is  figured  apertures 
in  M.  Fraunhofer's  work,  and  is  one  of  the  most  magnificent  phenomena  in  Optics.  l°r  gratings. 

When  we  look  at  a  bright  star  through  a  very  good  telescope  with  a  low  magnifying  power,  its  appearance  is      766. 
that  of  a  condensed,  brilliant  mass  of  light,  of  which  it  is  impossible  to  discern  the  shape  for  the  brightness  ;  Rings  seen 
and  which,  let  the  goodness  of  the  telescope  be  what  it  will,  is  seldom  free  from  some  small  ragged  appendages  about. tne 
or  rays.     But  when  we  apply  a  magnifying  power  from  200  to  300  or  400,  the  star  is  then  seen  (in  favourable  telescopes 
circumstances  of  tranquil  atmosphere,  uniform  temperature,  &c.)  as  a  perfectly  round,   well-defined  planetary 
disc,  surrounded  by  two,  three,  or  more  alternately  dark  and  bright  rings,  which,  if  examined  attentively,  are 
seen  to  be  slightly  coloured  at  their  borders.     They  succeed  each  other   nearly  at  equal   intervals  round  the 
central  disc,  and  are  usually  much  better  seen  and  more  regularly  and  perfectly  formed  in  refracting  than  in 
reflecting  telescopes.     The   central   disc,  too,  is  much   larger  in  the  former  than   in  the  latter  description    of 
telescope. 

These  discs  were  first  noticed  by  Sir  William  Herschel,  who  first  applied  sufficiently  high  magnifying  powers      767. 
to  telescopes  to  render  them  visible.     They  are  not  the  real  bodies  of  the  stars,  which  are  infinitely  too  remote  Spurious 
to  be  ever  visible  with  any  magnifiers  we  can  apply ;    but   spurious,  or  unreal  images,  resulting   from  optical  cllscs  of 
causes,  which  are  still  to  a  certain  degree  obscure.     It   is   evident,  indeed,   to  any  one  who  has  entered  into        stars' 
what  we  have  said  of  the  law  of  interferences,  and  from   the  explanation  given  in   Art.  590  and   591   of  the 
formation  of  foci  on  the  undulatory  system,  that  (supposing  the  mirror  or  object-glass  rigorously  aplanatic)  the 
focal  point  in  the  axis  will  be  agitated  with  the  united  undulations,  in  complete  accordance,  from  every  part  of 
the  surface,  and  must,  of  course,  appear  intensely  luminous  ;    but   that  as  we  recede  from   the  focus   in   any 
direction  in  a  plane  at  right  angles  to  the  axis,  this  accordance  will  no  longer  take  place,  but  the  rays  from  one 
side  of  the  object-glass  will  begin  to  interfere  with   and    destroy   those  from   the  other,  so  that  at  a  certain 
distance  the  opposition  will  be  total,  and  a  dark  ring  will  arise,  which,  for  the  same  reason,  will  be  succeeded 
by  a  bright  one,  and  so  on.     Thus  the  origin   both  of  the   central  disc  and  the  rings  is  obvious,  though  to  Explanation 
calculate  their  magnitude  from  the  data  may  be  difficult.     But  this  gives  no  account  of  one  of  the  most  remark-  of  the 
able  peculiarities  in  this  phenomenon,  viz.  that  the  apparent  size  of  the  disc  is  different  for  different  stars,  being  "n?s  on  the 
uniformly  larger  the  brighter  the  star.     This  cannot  be  a  mere  illusion  of  judgment ;  because  when  two  unequally  PnnclPI«  of 
bright  stars  are  seen  at  once,  as  in  the  case  of  a  close  double  star,  so  as  to  be  directly  compared,  the  inequality  j-"^~ 
of  their  spurious  diameters  is  striking  ;  nor  can  it  be  owing  to  any  real  difference  in  the  stars,  as  the  intervention 
of  a  cloud,  which  reduces  their  brightness,  reduces  also  their  apparent  discs  till  they  become  mere  points.     Nor 
can  it  be  attributed  to  irradiation,  or  propagation  of  the  impression  from  the  point  on  the  retina  to  a  distance,  as 
in  that  case  the  light  of  the  central  disc  would  encroach  on  the  rings,  and  obliterate  them  ;    unless,   indeed,  we 
suppose  the  vibrations  of  the  retina  to  be  performed  according  to  the  same  laws  as  those  of  the  ether,  and  to 
De  capable  of  interfering  with  them  ;    in  which  case,  the  disc  and  rings  seen  on  the  retina  will  be  a  resultant 
system,  originating  from  the  interference  of  both  species  of  undulations. 

Not  to  enter  further,  however,  on  this  very  delicate  question,  we  shall  content  ourselves  with  stating  some  of     ?*>8. 
the  phenonena  we  have  observed,  as  produced  by  diaphragms,  or  apertures  of  various   shapes  variously  applied  pllenomeni 
to  mirrors  and  object-glasses,  and  which  form  no  inapt  supplement  to  the  curious  observations  of  Fraunhofer  on  a 
the  effect  of  very  minute  apertures,  of  which  they  are  in  some  sort  the  converse.  various 

3  s  2  figures. 


492  LIGHT. 

Light.  When  the  whole  aperture  of  a  telescope  is  limited  by  a  circular  diaphragm,  whether  applied  near  to,  or  at  a 
V^^^B/  distance  from,  the  mirror  or  object-glass,  the  disc  and  rings  enlarge  in  the  inverse  proportion  of  the  diameter  of  ' 

769.  the  aperture.     When  the  aperture  was  much  reduced  (as  to  one  inch,  for  a  telescope  of  7  feet  focal  length)  the 
Circular       spurious  disc  was  enlarged  to  a  planetary  appearance,  being  well  defined,  and  surrounded   by  one   ring  only, 
apertures,     strong  enough  to  be  clearly  perceived,  and  faintly  tinged  with  colour  in  the  following  order,  reckoning  from  the 

centre  of  the  disc.     White,  very  faint  red,  black,  very  faint  blue,  white,  extremely  faint  red,  black.     When  the 
aperture  was  reduced  still  farther  (as  to  half  an  inch)  the  rings  were  too  4aint  to  be  seen,  and  the  disc  was  enlarged 
to  a  great  size,  the  graduation  of  light  from  its  centre  to  the  circumference  being  now  very  visible,  giving  it  a 
Fig.  152.      hazy  and  cometic  appearance,  as  in  fig.  152. 

770.  When  annular  apertures  were  used  the  phenomena  were  extremely  striking,  and  of  great  regularity.     The 
Annular       exterior  diameter  of  the  annulus  being    three   inches,  and   the  interior    1^,    the  appearance  of  Capella  was 

res"  as  in  fig.  153,  and  of  the  double  star  Castor,  as  in  154.  As  the  breadth  of  the  annulus  is  diminished,  the  size 
of  the  disc  and  breadth  of  the  rings  diminish  also,  (contrary  to  what  took  place  in  Fraunhofer's  experiments 
with  extremely  narrow  annuli,  and  obviously  referring  the  present  phenomena  to  different  principles,)  at  the  same 
1^  '  '°  time  the  number  of  visible  rings  increases.  Fig.  155,  156,  and  157  exhibit  the  appearance  of  Capella  with 
annular  apertures  of  5.5  inch  —  5  inch  (i.  e.  whose  exterior  diameter  =  5.5  and  interior  ==  5)  of  0.7  —  0.5,  of 
2.2  —  2.0.  In  the  last  case  the  disc  was  reduced  to  a  hardly  perceptible  round  point,  and  the  rings  were  so  close 
and  numerous  as  scarcely  to  admit  being  counted,  giving,  on  an  inattentive  view,  the  impression  of  a  mere 
circular  blot  of  light.  When  the  breadth  of  the  annulus  was  reduced  to  half  this  quantity,  the  intervals  between 
the  rings  could  no  longer  be  discerned.  The  dimensions  of  the  rings  and  disc,  generally,  seem  to  be  proportional 

r'  —  r 

to  . 

r 

77  j  Besides  the  rings  immediately  close  to  the  central  disc,  however,  others  of  much  greater  diameter  and  fainter 

Another  set  light,  like  halos,  are  seen  with  annular  apertures,  which  belong  (in  Fraunhofer's  language)  to  spectra  of  a 
of  rin«3.  different  class.  With  a  single  annulus  they  are  too  faint  to  be  distinctly  examined,  but  with  an  aperture 
Fig.  158  composed  of  two  annuli,  as  in  fig.  158,  they  are  very  distinct  and  striking,  presenting  the  phenomenon  in 

fig.  159,  (in  which  it  is  to  be  understood  that  light  is  represented  in  the  engraving  by  darknes-,  and  darkness 

by  light.) 

772.  When  the  aperture  was  in  the  form  of  an  equilateral  triangle,  the  phenomenon  was  extremely  beautiful ;   it 
Image  pro-  consisted   of  a   perfectly  regular,  brilliant,  six-rayed  star,  surrounding  a  well-defined   circular  disc  of  great 

;J  by  a  brightness,  Xhe  rays  do  not  unite  to  the  disc,  but  are  separated  from  it  by  a  black  ring.  They  are  very  narrow, 
aperture  an^  perfectly  straight ;  and  appear  particularly  distinct  in  consequence  of  the  total  destruction  of  all  the  diffused 

light  which  fills  the  field  when  no  diaphragm  is  used ;  a  remarkable  effect,  and  much  more  than  in  the  mere 
Fie  160  proportion  of  the  light  stopped.  Fig.  160  is  a  representation  of  this  elegant  appearance.  The  same  arises 

when,  in  place  of  an  equilateral  triangle,  the  aperture  is  the  difference  of  two  concentric,  equilateral  triangles 

similarly  situated. 

773.  A.S  a  triangle  has  but  three  side-  and  three  angles,  it  seems  singular  that  a  si.r-rayed  star  should  be  produced. 
When  out     Supposing  three  to   arise  from  the-  angles,  and  three  from  the  sides,  it  might  be  expected  that  some  sensible 
of  focus.      difference  should  exist  in  the  alternate  rays,  marking  their  different  origin.     When  the   telescope  is  in  perfect 

focus,  however,  all  the  rays  are  precisely  alike ;  but  if  thrown  out  of  focus,  their  difference  of  origin  becomes 
F'  161  apparent.  Fig.  161  represents  the  phenomenon  then  seen,  in  which  the  alternate  branches  are  seen  to  consist 
of  a  series  of  fringes  parallel  to  their  length,  and  the  others  of  small  arcs  of  similar  fringes  immediately  adjacent 
to  the  vertices  of  the  hyperbolas  to  which  they  belong,  and  which  consequently  cross  the  rays  in  a  direction 
perpendicular  to  their  length.  As  the  telescope  is  brought  better  in  focus,  the  hyperbolas  approach  their  asymp- 
totes, and  are  confounded  together  in  undistinguishable  proximity  ;  and  thus  three  rays  arise  composed  of  conti- 
nuous lines  of  light,  and  three  intermediate  ones  composed  of  an  infinite  number  of  discontinuous  points  placed 
infinitely  near  each  other.  To  represent  analytically  the  intensity  of  the  light  in  one  of  these  discontinuous  rays 
would  call  for  the  use  of  functions  of  a  very  singular  nature  and  delicate  management. 

774.  The  phenomenon  just  described  affords  in  certain  cases  a  very  perfect  position-micrometer  for  astronomical 
Application  uses.     If  the  diaphragm  be  turned  round,  the  rays  turn  with  it;  and  if  a  brilliant  star  (as  a  Aquilae)  have  near 
to  the  con-  jj  a  very  small  one,  the  diaphragm  may  be  so  placed  as  to  make  one  of  the  rays  pass  through  the  small  star, 

>f  which  thus  remains  like  a  bead  threaded  on  a  string,  and  may  be  examined  at  leisure.     If  then  the  position  of 
micrometer,  the  diaphragm  be  read  off  on  a  graduation  properly  contrived,  the   relative  situations  of  the  two  stars  become 
known.     We    have  satisfied  ourselves  by  trial   of  the  practicability  of  this  ;    and  by  proper  contrivances  the 
principle  may  be  made  available  in  cases  which  at  first  sight  appear  to  present  considerable  difficulties. 
775  When  three  circular  apertures,  having  their  centres  at  the  angles  of  an  equilateral  triangle,  were  used,  the 

Three '  '  image  consisted  of  a  bright  central  disc.  Six  fainter  ones  in  contact  with  it,  and  a  system  of  very  faint  halo- 
circular  like  rings  surrounding  the  whole  as  in  fig.  162.  When,  however,  three  equal  and  similar  annular  apertures 
apertures.  were  thus  disposed,  the  appearance  when  in  focus  was  as  in  fig.  153,  being  exactly  the  same  as  if  two  of  them 
Fig.  162.  were  closed.  But  when  thrown  a  little  out  of  focus,  the  difference  was  perceived.  Fig.  163  represents  the 
Fig.  163.  appearance  in  this  case,  each  of  the  apertures  then  produces  its  own  central  disc  and  system  of  rings,  whose 
intersections  give  rise  to  the  system  of  intersectional  fringes  there  depicted.  As  the  telescope  is  brought  better 
Fig.  164.  in  focus  these  disappear,  and  the  phenomenon  is  as  in  fig.  164  ;  the  centres  gradually  approaching,  and  the 

rings  blending  till  the  point  of  complete  coincidence  is  attained. 

_._  An  aperture  in  the  form  of  the  difference  between  two  concentric  squares  produced  not  an  eight,  but  a  four 

rayed  star.     The  rays,   however,  were  not,  as  in  the  case  of  the  triangular  aperture,  uninterrupted  fine  lines, 
gradually  tapering  away  from  the  centre  to  their  extremities,  but  composed  of  distinct  alternating  obscure  and 


LIGHT.  493 

Light.      bright  portions,  as  represented  in   fig.  165.     The  portions  nearest  the  central  disc  (which  is  circular)  were  Part  ilL 
— "\— ^  composed  of  bands  transverse  to  the  direction  of  the  rays,  and  tinged  with  prismatic  colour.     Similar  bands,  >——>,-•••' 
no  doubt,  existed  in  the  more  distant  portions,  which  extended  to  a  great  length. 

An  aperture  consisting  of  fifty  squares,  each  of  about  half  an  inch  in  the  side,  regularly  disposed  at  intervals  ^  "r^' 
so  as  to  leave  spaces  between  them  in  both  directions  equal  in  breadth  to  the  side  of  each,  produced  an  image      ^^^. 
totally  different  from  that  described    by  Fraunhofer  as   resulting  from  the  crossing  of  two  equal  very  close  Effect  of 
gratings,  though  the  distribution  and  shape  of  the  apertures  were  the  same  in  both  cases.     It  was  as  repre-  very  nume- 
sented  in  fig.  166,  consisting  of  a  white,  round,  central  disc,  surrounded  by  eight  vivid  spectra,  disposed  in  the  rous  square 
circumference  of  a  square,  beyond  which  were  arranged  in  the  shape  of  a  cross,  triple  lines  of  very  faint  spectra  p?er'jggS 
extending  to  a  great  distance. 

When  the  aperture  consisted  of  numerous  equilateral  triangles  regularly  disposed,  as  in  fig.  167,  the  image      778. 
presented  the  very  beautiful  phenomenon  represented  in  fig.  168,  consisting  of  a  series  of  circular  discs  arranged  Fig- 167. 
in  six  diverging  rays  from  the  central  one,  and  each  surrounded  with  a  ring.     The  central  disc  was  colourless  and 
bright ;   the  rest  more  and  more  strongly  coloured  and  elongated  into  spectra,  according  to  their  degree  of 
remoteness  from  the  centre.     These  are  only  a  few  of  the  curious  and  beautiful  phenomena  depending  on  the 
figures  of  the  apertures  of  telescopes,  which  afford  a  wide  field  of  further  inquiry,  and  one  at  least  as  interesting 
to  the  artist  as  to  the  philosopher. 


494 


L  I  G  H  T. 


Light. 


PART  IV. 


OF  THE  AFFECTIONS  OF  POLARIZED  LIGHT. 


§  I.   Of  Double  Refraction. 


779. 

Exceptions 
to  the  law 
of  ordinary 
refraction 
numerous. 


Classes  of 
bodies  in 
which  it 
holds. 


780. 
Double 
refraction. 


WHEN  a  ray  of  light  is  incident  on  the  surface  of  a  transparent  medium,  a  portion  of  it  is  reflected,  at  an 
angle  equal  to  that  of  incidence,  another  small  portion  (-so  small,  however,  that  we  shall  neglect  its  consi- 
deration) is  dispersed  in  all  directions,  serving  to  render  the  surface  visible,  and  the  rest  enters  the  medium  and 
is  refracted.  The  law  of  refraction,  or  the  rule  which  regulates  the  path  of  this  portion  within  the  medium, 
has  been  explained  in  the  preceding  parts  ;  and  no  exceptions  to  it,  as  a  general  law,  have  hitherto  been  noticed. 
It  is,  however,  very  far  from  general ;  and,  in  fact,  obtains  only  where  the  refracting  medium  belongs  to  one  or 
other  of  the  following  classes,  viz. 

Class  1.   Gases  and  vapours. 

2.  Fluids. 

3.  Bodies  solidified  from  the  fluid  state  too  suddenly  to  allow  of  the  regular  crystalline  arrangement  of 

their  particles,  such  as  glass,  jellies,  &c.,  gums,  resins,  &c.,  being  chiefly  such  as  in  the  act  of 
cooling  pass  through  the  viscous  state. 

4.  Crystallized  bodies,  having  the  cube,  the  regular  octohedron,  or  the  rhomboidal  dodecahedron  for 

their  primitive  form,  or  which  belong  to  the  tessular  system  of  Mohs.  A  very  few  exceptions 
(probably  only  apparent  ones,  arising  from  our  imperfect  knowledge  of  crystallography)  exist  to 
the  generality  of  this  class. 

The  solid  bodies  belonging  to  these  classes,  moreover,  cease  to  belong  to  them  when  forcibly  compressed  or 
dilated,  either  by  mechanical  violence,  or  by  the  unequal  action  of  heat  or  cold,  which  brings  their  particles 
into  a  state  of  strain,  such  as  in  extreme  cases  to  produce  their  disruption,  as  is  familiarly  seen  in  the  cracking 
of  a  piece  of  glass  by  heat  too  suddenly  and  partially  applied.  The  cla~s  of  fluids  too  admits  some  exceptions, 
at  least  when  very  minutely  considered ;  but  the  deviation  from  the  ordinary  law  of  refraction  in  these  cases  is 
of  so  microscopic  a  kind,  that  we  shall  at  present  neglect  to  regard  it. 

All  other  bodies,  comprehending  all  crystallized  media,  such  as  salts,  gems,  and  crystallized  minerals,  not 
belonging  to  the  system  above  mentioned  ;  all  animal  and  vegetable  bodies  in  which  there  is  any  disposition  to 
a  regular  arrangement  of  molecules,  such  as  horn,  mother  of  pearl,  quill,  &c. ;  and,  in  general,  all  solids  when 
in  a  state  of  unequal  compression  or  dilatation,  act  on  the  intromitted  light  according  to  very  different  laws, 
dividing  the  refracted  portion  into  two  distinct  pencils,  each  of  which  pursues  a  rectilinear  course  so  long  as  it 
continues  within  the  medium,  according  to  its  own  peculiar  laws,  but  without  further  subdivision.  This  pheno- 
menon is  termed  double  refraction.  It  is  best  and  most  familiarly  seen  in  the  mineral  termed  Iceland  spar, 
which  is,  in  fact,  carbonate  of  lime  in  a  regular  crystalline  form.  This  is  generally  obtained  in  oblique  parallel- 
epipeds, easily  reduced  by  cleavage  to  regular,  obtuse  rhomboids,  and  is  not  uncommonly  met  with  in  a  state  of 
limpid  transparency,  on  which  account,  as  well  as  by  reason  of  its  remarkable  optical  properties,  it  easily 
attracted  attention.  Bartholinus,  in  1669,  appears  to  have  been  the  first  to  give  any  account  of  its  double 
refraction,  which  was  afterwards  more  minutely  examined  by  Huygens,  the  first  proposer  of  the  undulatory 
theory  of  light,  whose  researches  on  this  phenomenon  form  an  epoch  in  the  history  of  Physical  Optics  little  if 
at  all  less  important  than  the  great  discovery  of  the  different  refraiigibility  of  the  coloured  rays  by  Newton.  To 
Huygens  we  owe  the  discovery  of  the  law  of  double  refraction  in  this  species  of  medium.  Newton,  misled  by 
some  inaccurate  measurements,  (a  thing  most  unusual  with  him,)  proposed  a  different  one  ;  but  the  conclusions 
of  Huygens,  long  and  unaccountably  lost  sight  of,  were  at  length  established  by  unequivocal  experiments  by 
Dr.  Wollaston,  since  which  time  a  new  impulse  has  been  given  to  this  department  of  Optics ;  and  the  successive 
labours  of  Laplace.  Malus,  Brewster,  Biot,  Arago,  and  Fresnel  present  a  picture  of  emulous  and  successful 
research,  than  which  nothing  prouder  has  adorned  the  annals  of  physical  science  since  the  developement  of  the 
true  system  of  the  universe.  To  enter,  however,  into  the  history  of  these  discoveries,  or  to  assign  the  share  of 
honour  which  each  illustrious  labourer  has  reaped  in  this  ample  field  forms  no  part  of  our  plan.  Of  the  splendid 
constellation  of  great  names  just  enumerated,  we  admire  the  living  and  revere  the  dead  far  too  warmly  and  too 
deeply  to  suffer  us  to  sit  in  judgment  on  their  respective  claims  to  priority  in  this  or  that  particular  discovery ; 
to  balance  the  mathematical  skill  of  one  against  the  experimental  dexterity  of  another,  or  the  philosophical 
acumen  of  a  third.  So  long  as  "  one  star  differs  from  another  in  glory,'' — so  long  as  there  shall  exist 
varieties,  or  even  incompatibilities  of  excellence, — so  long  will  the  admiration  of  mankind  be  found  sufficient 
for  all  who  truly  merit  it.  Waving,  then,  all  reference  to  the  history  of  the  subject,  except  in  the  way  of  inci- 
dental remark,  or  where  the  necessity  of  the  case  renders  it  unavoidable,  we  shall  present  the  reader  with  as 


L  I  G  II  T.  495 

Light,     systematic  an  account  as  we  are  able,  of  the  present  state  of  knowledge  with  respect  to  the  laws  and  theory  of    Part  IV. 
•"•v^*1  Double  Refraction.     The  Huygenian  law  having  been  demonstrated  to  apply  rigorously  to  the  case  for  which  v-» -v~™-' 
he  himself  devised  it,  as  well  as  to  a  very  large  class  of  other  bodies,  we  shall  begin  with  that  class,  and  proceed 
afterwards  to  consider  more  complicated  cases. 

In  all  crystallized  bodies,  then,  which  possess  double  refraction,  it  is  found  that  that  portion  of  a  ray  of      78], 
ordinary  light  incident  on  any  natural  or  artificially  polished  surface  which  enters  the  body  is  separated  into  two  Axes  of 
equal  pencils  which  pursue  rectilinear  paths,  making  with  each  other  an  angle  not  of  constant  magnitude,  but  doubl< 
varying  according  to  the  position  which  the  incident  ray  holds  with  respect  to  the  surface,  and  to  certain  fixed  re 
lines,  or  axes  within  the  crystal,  and  which  lines  are  related  in  an  invariable  manner  to  the  planes  of  cleavage, 
or  other  fixed  planes  or  lines  in  the  primitive  form  of  the  crystal.     Now,  it  is  found  that  in  every  crystal  there 
is  at  least  one  such  fixed  line,  along  which  if  one  of  these  two  pencils  be  transmitted  the  other  is  so  also,  so 
that  in  this  case  the  two  pencils  coincide,  the  angle  between  them  vanishing.     Moreover,  no  crystal  has  yet  been 
discovered  in  which  more  than  two  such  lines  exist.     These  lines  are  called  the  optic  axes.     All  double  refracting 
crystals,  then,  at  present,  may  be  divided  into  such  as  have  one,  and  such  as  have  two,  optic  axes. 

When  a  ray  penetrates  the  surface  of  a  crystal  so   as  to  be  transmitted  undivided  along   the    optic  axis ;       792. 
or  when,  moving  within  the  crystal  along   that   line,  it  meets    the  surface   and  passes   out,   whatever  be  the  Rayf 
inclination    of  the   surface,  its   refraction  is  always  performed  according   to  the  ordinary  law  of  the  proper-  m. 
tional  sines.     Thus,  in  this  particular  case,  the  crystal  acts  precisely  as  an  uncrystallized  medium,  (some  rare  axesbsuffer 
instances  excepted,  of  which  more  hereafter.)  ordinary 

But  in  all  other  cases  the  law  is  essentially  different,  and   (for  one  portion  of  the  divided  pencil,  at  least)  refraction 
of  a   very  singular  and  complicated  nature.     This  we   shall  first  proceed    to  explain    in  the  simpler  case  of  °nl;J.'So 
crystals  with    one   optic   axis.      But,  first,   we  must   explain  somewhat  more  distinctly,  what  we  mean    by  wh     • 
axes  and  fixed  lines  within  a  crystal.     Suppose  a  mass  of  brickwork,  or  masonry,  of  great  magnitude,  built  of  meant  "- 
bricks,  all  laid  parallel  to  each  other.     Its  exterior  form  may  be  what  we  please ;  a  cube,  a  pyramid,  or  any  other  axes  and 
figure.     We  may  cut  it  (when  hardened  into  a  compact  mass)  into  any  shape,  a  sphere,  a  cone,  or  cylinder,  &c. ;  fixed  lines 
but  the  edges  of  the  bricks  within  it  lie  still  parallel  to  each  other;  and  their  directions,  as  well  as  those  of  the  witl"n  a 
diagonals  of  their  surfaces,  or  of  their  solid  figures,  may  all  be  regarded  as  so  many  axes,  i.  e.  lines  having  (so  cr^sta '• 
long  as  the  mass  remains  at  rest)  a  determinate  position,  or   rather  direction  in  space,  no  way  related  to  the 
exterior  surfaces,  or  linear  boundaries  of  the  mass,  which  may  cut  across  the  edges  of  the  bricks  in  any  angles 
we  please.     Whenever,  then,  we  speak  of  fixed  lines,  or  axes  of,  or  within,  a  crystal,  we  always  mean  directions 
in  space  parallel  to  each  of  a  system  of  lines  drawn  in  the  several  elementary  molecules  of  the  crystal,  according 
to  given  geometrical  laws,  and  related  in  a  given  manner  to  the  sides  and  angles  of  the  molecules  themselves. 
We  must  conceive  the  axis,  then,  of  a  crystallized  mass  not  as  a  single  line  having  a  given  place,  but  as  any  line 
whatever  having  a  given  direction  in  space,  i.  e.  parallel  to  the  axis  of  each  molecule,  which  is  a  line  having  a 
determinate  place  and  position  within  it. 

In  the  remainder  of  this  section,  when  we  speak  of  the  axis  or  axes  of  a  crystallized  mass  or  surface  generally,       784. 
we  mean  the  direction  of  the  optic  axis  or  axes  of  its  molecules,  or  of  a  crystal  similar  and  similarly  situated 
1o  any  one  of  them. 

Of  the  Law  of  Double  Refraction  in  Crystals  with  One  Optic  Axis. 

This  class  of  crystals  comprises  all  such  as  belong  to  Mohs's  rhombohedral  system,  or  which  have  the  acute  or      7S5. 
obtuse  rhomboid,  or  regular  six-sided  prism,  for  their  primitive  form,  as  well  as  all   which   belong   to  his  Enumera- 
pyramidal  system,  or  whose  primitive  form  is  either  the  octohedron  with  a  square  base,  the  right  prism  with  a  tionofcrys- 
square  base,  or  the  bi-pyramidal  dodecahedron.     All  such  crystals  Dr.  Brewster  has  shown   to  have  but  one  tals.  hav'"o 
axis,  which  is  that  to  which  the  primitive  form  is  symmetrical,  viz.  in  the  rhomboid,  the  axis  of  the  figure,  or  gXS;'snfne 
line  joining  the  two  angles  formed  by  three  equal  plane  angles ;  in  the  hexagonal  prism,  the  geometrical  axis  classes. 
of  the  prism  ;  in  the  octohedron,  or  square  based  prism,  a  line  drawn  through  the  centre  of  the  base  at  right 
angles  to  it.     The  cases  in  accordance  with  the  rule  are  so  numerous,  and  the  exceptions,  once  believed  to  be 
so,  have  so  often  disappeared  on  the  attainment  of  a  more  perfect  knowledge   of  the  crystalline  forms  of  the 
excepted  minerals,  that  when  any  case  of  disagreement  seems  to  occur,  we  are  justified  in  attributing  it  rather 
to  our  own  incorrect  determination  of  this  datum,  than  to  want  of  generality  in  the  rule  itself. 

In  all  crystals  of  this  class,  one  of  the  two  equal  pencils  into  which  the  refracted  ray  is  divided  follows  the      786. 
ordinary  law  of  Snellius  and  Descartes,  having  a  constant  index  of  refraction  (/t),  or  invariable  ratio  of  the  sine  Refraction 
of  incidence  to  that  of  refraction,  whatever  be  the  inclination  of  the  surface  by  which  it  enters ;   so  that  its  of  tlle  °™!~ 
velocity  within  the  medium,  when  once  entered,  is  the  same  in  whatever  direction  it  traverses  the  molecules ;  Ij*-ry  ^  "L 
and  with  respect  to  this  ray  the  crystal  comports  itself  as  an  uncrystallized  medium.     This,  then,  is  called  the  crystal*!*  ' 
ordinary  pencil. 

To  understand  the  law  obeyed  by  the  other,  or  extraordinary  portion  of  the  divided  pencil,  let  us  consider      707 
it  as  fairly  immersed  in  the  medium,  and  pursuing  its  course  among  the  molecules.     Then  its  velocity  will  not,  Huyeens's 
as  in  the  case  of  the  ordinary  ray,  be  the  same  in  whatever  direction  it  traverses  them,  but  will  depend  on  the  law  for  the 
angle  it  makes  with  the  axis ;   being  a  minimum  when  its  path  within  the  crystal  is  parallel  to  the  axis,  and  a  velocity  of 
maximum  when  at  right  angles  to  it,  or  vice  versa  ;   and  in  all  intermediate  inclinations   of  an  intermediate  the.extri>- 
magnitude  according  to  the  following  law.     Let  an  ellipsoid  of  revolution,  either  oblate  or  prolate,  as  the  case  ™'"ary 


496  LIGHT. 

Light.      may  be,  be  conceived,  having  its  axis  of  revolution  coincident  in  direction  with  the  axis  of  the  crystal,  and  its  polar     Part  'v 
v— •v'*''  to  its  equatorial  radius  in  the  ratio  of  the  minimum  and  maximum  velocities  above  mentioned,  i.  e.  as  the  velocity 
of  a  ray  moving  parallel  to  that  of  one  perpendicular  to  the  axis.     Then  in  all  intermediate  positions,  the  radius 
of  this  spheroid  parallel  to  the  ray  will   represent  its  velocity  on  the  same  scale  that  its  polar  and  equatorial 
radii  represent  the  velocities  in  their  respective  directions. 

788.  This  is  the  Huygenian  law  of  velocities,  in  its  most  simple  and  general  form.     It  does  not  at  first  sight  appear 
Its  con-        what  this  has  to  do  with  the  law  of  extraordinary  refraction  ;  but  the  reader  who  has  considered  with  the  requisite 
nection  with  attention  what  has  been  said  in  Art.  539,  540,  with  prospective  reference  to  this  very  case,  will  easily  perceive 

aw,.       that,  the  law  of  velocity  of  the  ray  within  the  medium  once  established,  it  becomes  a  mere  matter  of  pure 

nary  refrac-  Geometry  to  deduce  from  it  the  law  of  extraordinary  refraction,  whether  we  adopt  the  Corpuscular  theory,  and 

tiori.  employ  Laplace's  principle  of  least  action,  as  in  that  Article  ;  or  whether,  preferring  the  Undulatory  hypothesis, 

we  substitute  for  this  principle  the  equivalent  one  of  swiftest  propagation,  as  explained  in  Art.  587,  588.     We 

should  observe,  however,  that  the  Huygenian  law,  as  just  stated,  is  worded  in  conformity  with  the   undulatory 

doctrine,  in  which  the  velocity  in  a  denser  medium  is  supposed  slower  than  in  a  rarer.     But  when  we  use  the 

principle  of  least  action,  we  must  invert  the  use  of  the  word,  or,  which  comes  to  the  same  thing,  suppose  the 

the  velocity  in  the  medium   to  be  inversely  proportional   to  the   radius  of  the   ellipsoid.     The  results  being 

necessarily  the  same  in  both  cases,  we  shall  use  at  present  the  language  of  the  Corpuscular  system. 

789.  Retaining,  then,  the  notation  of  Art.  540,  the  law  of  refraction  will  be  derived  from  the  equation  V  .  S  -(-  V .  S ' 
Investigi-     __  a  m;nimum>  where  V  is  the  velocity  without,  and  V  that  within  the  medium,  and  where  S  and  S'  are  the  spaces 
latter  from    described  without  and  within  it,  in  the  passage   of  a  ray  from  point  to  point.     Let  a  and  b  be  the  polar  and 
the  former     equatorial  semiaxes  of  the  ellipsoid  above  spoken  of,  (which  we  shall  call  the  ellipsoid  of  double  refraction,)  and 
la»              let  a,  /3,  «y  be  the  coordinates  of  the  point  (A)   without  the  crystal,  and  a',  ft',  7'  those  of  one  (B)  within  it, 

through  which  the  ray  is  supposed  to  pass,  and  x,  y,  z  the  coordinates  of  a  point  in  the  surface  of  the  crystal,  on 
which  it  must  be  incident,  so  as  to  be  capable  of  passing  from  A  to  B  in  the  manner  required  by  the  law  of 
extraordinary  refraction  ;  and  let  0  be  the  angle  which  the  interior  portion  S'  makes  with  the  axis  of  the  crystal. 

Expression   Then  will  the  radius  of  the  spheroid  parallel  to  this  portion  (by  conic  sections)  be  expressed  by 

for  the 

radius  of  the  a  J  a  J 

spheroid  of  r  =  =  -rrr-  ;    (1) 

refraction.  V  ft2  .  sin  0'J  +  a?  COS  02 

where  a  is  the  equatorial,  and  6  the  polar  radius  of  the  spheroid.  Now,  if  we  take  p.  to  represent  the  index  of 
ordinary  refraction,  since  we  have,  generally,  V  =  —  — ,  and  since,  when  r  =  6  the  extraordinary  and  ordinary 

rays  coincide,  and   therefore  V  —p.  V,  consequently  we  must  have  p,  V  =  — — ,  and  const  =  b  p  V,  so  that 

\) 

we  shall  get 

/_  b  V'  h 

T    *  "V  f 

7o,Q  In  general,  as  we  have  already  seen,  the  condition  of  least  action  affords  the  equation 

Introduc*.  d  {  V  S  +  V  S'  }  =  0,  or  V  .  d  S  +  V .  d  S'  +  S' .  d  V  =  0  ;     (2) 

principle  of   ^ut  to  ma'ie  use  °^  tn's>  we  must  express  V,  S,  and  S',  in  terms  of  variable  quantities  relating  to  a  point  any 

least  action   how  taken  in  the  surface  of  the  crystal.     Whether  this  point  be  expressed  by   rectangular  or  polar  coordinates 

or  swiftest     is  no  matter:  it  will  be  more  convenient,  however,  to  use  polar.     Let,  then,  C  (fig.  169)  be  the  point  of  inci- 

propagation.  dence  of  the  ray  A  C  on  the  surface  H  a  O  i,  and  about  C  as  a  centre  describe  a  sphere.     Let  Z  C  2  be  the  per- 

Fig.  169.      pendicular  to  the  surface  at  C,  and  let  P  Cp  be  the  position  of  the  axis  of  the  crystal.     The  plane  Z  P  H  z  p  O  Z 

perpendicular  to  the  surface,  and  passing  through  the  axis,  is  called  the  principal   section  of  the  surface.     Let 

Z  A  a,  z  B  b  be  vertical  planes,  containing  the  incident   and  refracted  rays,  and  join  B  p  by  the  arc  of  a  great 

circle.     Then  it  is  evident,  that  this  arc  will  be  equal  to  0. 

791.  Suppose,  now,  the  axis  of  the  x  to  be  parallel  to  H  C  the  projection  of  the  axis  of  the  crystal,  and  since  we  may 

choose  the  plane  of  the  x,  y,  as  we  please,  let  it  coincide  with  the  refracting  surface,  so  that  z  =  0.  Then 
dropping  the  perpendiculars  A  M,  M  m,  B  N,  N  n,  and  putting  \  =  ZP=zp=  angle  between  the  axis  and 
perpendiculars. 

•ss  —  O  a  =  inclination  of  the  plane  of  incidence  to  the  principal  section. 
•as'  =  O  b  =  inclination  of  plane  of  refraction  to  ditto. 
0  =  angle  Z  C  A  =  Z  A  =  angle  of  incidence 
0'  =  z  C  B  =  z  B  =  angle  of  refraction. 
We  shall  have  as  follows: 

AC  =  S  ;  AM  =  7  ;  C  m5  =  (a  -  *)' ;  M  ms  =  (j3  -  y)«  ; 
consequently, 

( 

a  —  x  =  7  .  tan  0  .  cos  w :   B  —  y  =  -v  .  tan  0  .  sin  OT  ;  S  =  -    -= 

cos  0  r 

and,  similarly,  \          (3) 


o'  -  x  =  7' .  tan  ff .  cos  -a';  ft'  -  y  =  */'  .  tan  &  .  sin  -a'  \  S'  =  -^— ; 


LIGHT.  197 

Lizht.      Now,  differentiating-  these  equations,  and  considering  that  d  (a  -  x)  =  d  (a'  -  x)  and  d  (/3  -  y)  =  d  (/31  -  y>     Part  IV. 
•— v-™""'  we  get  s— •v""1* 

d  (tan  0  .  cos  TO)  =  —  .  d  (tan  0' .  cos  TO')  ; 
7 

/ 

d  (tan  0  .  sin  TO)  =  —  .  d  (tan  0'  .  sin  TO')  ; 

7 
which  equations,  developed  and  reduced,  afford  the  following, 

d  0         7'    /cos  0V  ,       d  0         7'  _  .,  ,  ,, 

d  0'  ~   <u   '  Icos  0') '  °'  W        d  TO'  ~~    7  I 

>  (4) 

d  OT         7'      sin  (TO'  —  TO)  d  TO        7'    tan  0'  , 

=  — -    i :  =:  -—  . cos  (TO    —  TO)  ; 

d  0'        7      tan  0  .  cos  0'2  d «'      '  f     tan  0 

which  are  necessary  conditions,  in  order  that  the  point  C  may  remain  on  the  surface. 

But  since  S,  S',  V  may  be  regarded  as  functions  of  0'  and  TO',  which  are  the  polar  coordinates  we  propose  to        792. 
use  as  independent  variables,  we  shall  have 


aiid,  moreover, 


ds  =  1^^(d0         +d0_        \ds,=  -/_^!dfft 
cos  02    \d  0  '   d  ia'        /  cos  v* 


so  that,  substituting  their  values  in  the  equation  (2,)  we  get 

•  sin  °   d  l 


-     v 

" 


—  -  -         — 

cos  (?2    •  d  ff'  ^  cos  £>'*     r  cos  0'  '   d  ff 

.  sin0     d0  7'        d 


in  which  the  coefficients  of  each  of  the  two  independent  differentials  being  separately  made  to  vanish,  we  get 

1I'=  _  V  .  JL  .  ™£l£2i£  .  !»   _  V'.tan*'   ^ 
d  0'  7'  cos  0a-  d  0' 

d_V' _  7        sin  0  .  cos  0'      d0  | 

Ct  'CT'  <v  COS  U^  C*  7tf 

ft  f)  rf  f) 

In  these,  substituting  the  values  of  -—  and  -j-^  found  in  equation  (4,)  we  obtain  the  following 


dV  ' 

-r-;  =  —  V  .  sin  0  .  sin  0'  .  sin  (CT  - 


These  are  the  very  same  equations  with  those  deduced  by  Laplace  and  Malus,  by  a  more  abstruse  and  compli- 
cated calculus,  from  the  primary  dynamical  relations  of  the  problem,  and  from  them  it  is  easy  to  express,  in 
general,  the  law  of  refraction  corresponding  to  any  given  law  of  velocities,  for  we  have  only  to  put  them  under 
the  form 

d  V/ 

V  .  sin  0  .  cos  w    cos  «'  -f-  V  .  sin  0  .  sin  w  .  sin  -of  =  —  V  .  sin  ff  —  cos  0  .  -j-r, 

I      dV 

V  .  sin  0  .  cos  w  .  sin  w'  —  V  .  sin  0  .  sin  OT  .  cos  "sf  =  —  —  -r  -  —  .  ; 

sin  0    d-ss 

and  multiplying  the  first  by  cos  OT',  and  the  second  by  sin  fa',  and  adding,  we  get 

sin  CT'      d  V  ,   d  V  .   Tr 

V  .  sin  0  .  cos  «j  =  —  —  -  .  -  —  -  —  cos  0'  .  cos  -a/  .  —r~,  —  sin  &  .  cos  •a  .  V  ; 
'        ' 


—  —  -  .  -  —  -  —  .  .  —r~ 

sm  0'     d'Sf  do 


,.  . 
(b) 


COSCT'  dV  ,    dV 

V  .  sin  0  .  sm  w  =  --  :  —  -.  -?—.  -  cos  0'  .  sin  vs  .  -—-.  -  sin  9  .  sm  w  .  V  ;  (9) 

sin  fr  aw  d  0' 

VOL.  IV  3l 


and,  again,  multiplying  the  first  by  sin  ro',  and  the  second  by  —  cos  vs',  and  adding,  we  find 


498  LIGHT. 

Light.      Now,  the  second  members  of  these  equations,  (when  V  the  velocity  of  the  extraordinary  ray  is  any  function  of  0    Fan  IV, 

'  the  angle  it  makes  with  the  axis,  or  of  its  position  within  the  crystal,)  is  always  explicitly  given  in  terms  of  0'  s^-v— «•- 
and  •m',  so  that,  calling  P  and  Q  their  values  so  expressed,  we  have  at  once 


Q 


tan  w  =  — -  ;  cos  TS  = -;  sin  0  =  A/  P*  +  Q*  ; 

V  pi  _|_  Q* 

so  that  OT  and  6  are  directly  expressed  in  terms  of  -a'  and  0' ;  and,  therefore,  the  direction  in  which  a  ray,  moving 
anyhow  within  the  crystal  will  emerge,  is  known,  and  vice  versa. 

It  only  remains  to  execute  these  processes  in  the  case  before  us.     To  this  end  (for  simplicity)  we  shall  put 

V  =  1,  and  suppose  (since  a  and  6,  the  semiaxes  of  the  spheroid,  are  arbitrary)  6  =  —  ,  or  /t  =  — ,  and  put  VV 

for  the  radical  v'  a2 .  cos  0a  +  b*  .  sin  <ft1,  when  we  shall  have 

W  a*-b*      cos0 

V   =T6;dV  =  -T6-  v-^ 

Now  in  the  spherical  triangle  Z  B  p  we  have,  the  side  Z  p  =  X  ;  Z  B  =  ff,  angle  p  Z  B  =r  itf,  and  side/;  B  =  0, 
therefore,  by  spherical  trigonometry, 

cos  0  =  cos  X .  cos  ff  -j-  sin  X  .  sin  6' .  cos  ro',  (10) 

and  differentiating  separately  with  respect  to  6'  and  -a', 

d  .  cos  0 


dff 


:=  —  cos  X  .  sin  ff  +  sin  X  .  cos  ff  .  cos  w' 

d  .  cos  0  .       , 

---  ;  —  =  —  smX  .  sin  0   .  sin  sr. 
d  -a 

If,  then,  we  write  these  values  in  the  partial  differences  of  V  in  the  equations  (8)  and  (9,)  they  will  become 
sin0.  cos  w  =  --  T~vyl  W2.  sin^'.  cosra'-4-(a5  —  6°)  cos  0  [sinX  (1  -  cos  us".  sin  &•)  —  cos  X.  sin  0'.  cos  0'.  cos  si*]  [ 

sin#.  sin  w=  --  —  <  Ws.sin0'.  sinw'  —  (a8  —  ft*)  cos  0  [sin  X.  sin  sj'.cossj'.sinO'2  -f  cos  X.  sin  &  .  cos  &  .  sin  •a'}  !- 

W  t/   W  I 

In  these,  let  ft*  +  (a*  —  6a)  cos  0*  be  put  for  W8,  and,  bearing  in  mind  that  the  value  of  cos  0  is  as  given  in 
the  equation  (10,)  we  shall  see  that  they  will  reduce  themselves  respectively  to 

sin  0  .  cos  -a  =  —  -j  6a  .  sin  ff  .  cos  w'  +  (a9  —  ft5)  .  sin  X  .  cos  0  ? 

that  is,  by  reason  of  (10,) 

-  b-}  .  cos  X  .  sin  X  .  cos  ff  -f  (aq  .  sin  X2  -f-  b-  .  cos  X2)  .  cos  -a'  .  sin  P 


-  sin  6  .  cos  w  — 


a  6  W 

and  V,         (1  Q 

.  sin  0'  .  sin  OT' 


—  sine.  sinw=- 


abVf 


794.          These  equations,  conjointly  with  the  equations  expressing  the  value  of  W  in  terms  of  cos  0,  and  of  cos  0  in 
terms  of  ff  and  ro',  afford  a  complete  solution  of  the  problem  in  the  case  when  a  ray  passes  out  of  a  crystal  into 
air,  and  suffice  to  determine  both  the  inclination  of  the  refracted  ray  to  the  surface,  and  the  inclination  of  the 
plane  in  which  it  lies  to  the  principal  section. 
For  brevity,  let  us  put 

a8  .  sin  Xs  +  6«  .  cos  X«  =  A;     a",  cos  X*  +  62.  sinX*  =  B  ;    (a8  -  62)  .  sin  X.  cos  X  =  C;         (12) 

and,  dividing  the  second  of  the  equations  (11)  by  the  first,  we  find 

6'.  tanfl'.sinro' 

- 


which  gives  immediately  the  inclination  of  the  plane  of  emergence  to  the  principal  section,  or,  as  it  is  sometimes 
termed,  the  azimuth  of  the  emergent  ray. 


LIGHT.  499 

Reciprocally,  if  having  given  the  angle  of  incidence  and  azimuth  of  a  ray  incident  externally  on  the  crystal,     Pit  IV. 
'  we  would  find  the  angle  of  refraction  and  azimuth  of  the  intromitted  ray,  we  must  find  &  and  OT'  from  the  above  V— *v— ' 
equations  in  terms  of  6  and  w.     This  may  be  thus  accomplished  :  795. 

Given  the 

Take  x  —  tan  0' ,  cos  CT',  and  y  =  tan  ff  .  sin  vf,  path  of  the 

ray  without, 

then  J?  +  yl  =  tan  0",  and  cos  0*  =  - ;  thTwUhin 

1  -f  •*•  +  V  the  crystal. 

> 
and,  moreover,  tan  -a  — 


Ax  +  C' 
now,  since  W8  =  b'  +  (a8  —  6")  .  cos  0«, 

=  cos  0'*  \ —  +  (a8  -  6s)  (cos  X  +  sin  X  .  tan  &  .  cos  w')2  j- 

L  cos  0  *  j 

the  second  of  the  equations  (11)  becomes,  by  squaring, 

{b*  "")        6s 
-T-  +  (a8- ft8).  (cosX  +  sin X  .  tan  0' .  cos  w')s  5-=  —  (tan0'.  sin  TO')*, 
cos  0"                                                                        j        as 

that  is, 

a8  (sin  0  .  sin  w)8  {  68  (1  +  <r8  +  y8)  +  (a8  -  62)  (cos  \  +  x  .  sin  X)2  }  =  b2  y\ 

that  is,  developing 

a- .  (sin  0  .  sin  ro)2  {  A  x2  +  2  C  x  +  B  +  62  y-  }   =  b2  y2. 

b2y  62  y  C 

Now  we  have  A  x  +  C  =  •,  and  *  = — . 

tan  TO  A .  tan  TO        A 

And,  on  substitution,  this  equation  will  be  found  to  take  the  form  p  y2  +  q  =  0,  and  being  resolved  to  give 

a? .  sin  0  .  sin  TS 
y  =  tan  &  .  sin  TO'  =  —  ,  (14) 

VA  —  a2  .  sin  02  (A  .  sin  TO2  -j-  62  .  cos  sr2) 
and  substituting  this  in  the  value  of  x,  we  find 

a2  b2                                   sin  0  .  cos  TO                                    C 
x  =  tan  &  .  cos  TO'  =  — — -  . — .  (15) 

v  A  —  a2  .  sin  6>2  {  A  .  sin  TO2  +  62 .  cos  TO2  }        A 

These  equations  are  identical  with  those  demonstrated  by  Malus  in  his  Theorie  de  la  Double  Refraction,  with  some 
slight  differences  of  notation  only,  arising  from  our  having  reckoned  TO  and  w7  from  the  opposite  point  of  the  circle. 

The  values  of  A,  B,  and  C  depend  only  on  a,  b,  and  X,  that  is,  on  the  peculiar  nature  of  the  crystal,  which       796. 
determines  the  ratio  of  the  axes  of  the  spheroid  of  double  refraction,  and  on  the  inclination  of  the  axis  to  the  Particular 
surface  on  which  the  ray  is  incident.     The  former  are  constant  for  one  and  the  same  crystal,  however  the  surface  applications, 
be  placed  ;  the  latter  is  constant  for  any  given  surface.     Hence  it  appears,  that  the  general  law  of  extraordinary 
refraction,  when  we  confine  ourselves  to  the  consideration  of  a  surface  given  in   position  with  respect  to   the 
axis,  resolves  itself  into  an  infinite  variety  of  particular  laws,  some  of  which  we  shall  now  consider. 

Case  1.     X  =  0,  the  surface  perpendicular  to  the  axis  ;  A  =  62 ;  B  =  a*  ;  C  =  0,  and  the  equations  (14)  and      797. 
(15)  become  1st.  When 

a2         sin  0 .  sin  TO  a2          sin  0 .  cos  -a 

tan  0'  .  sin  TO'  =  —  .  —  —  ;  tan  0'  .  cos  TO'  =  —  .  —  — - ;  perpendici 

surface. 

these  equations  (as  well  as  Equation  13)  give  TO'  =:  TO,  so  that  in  this  case  the  plane  of  refraction  is  the  same  with 
that  of  incidence,  and  the  extraordinary  ray  is  not  deviated  out  of  the  vertical  plane.     Hence,  we  get  simply 

a8  sin  9 

tan  &==—.  ;  (16) 

6        Vl  -  a2 .  sin  0- 

which  expresses  the  law  of  extraordinary  refraction  in  this  case.     If  0  =  0,  &  =  0,  or  the  ray  incident  perpen- 

ct8  1 

dicularly  passes  unrefracted  along  the  axis.     If  0  =  90°,  tan  0'  = rr     =.      Now  if  we  put  6  =  —  and 

b  V  1  _  0«  /» 

a  =  — ,  this  becomes 

tan0'=  ===r;  (17) 

f!  J  ju'2  —  1 

which,  ft  and  n'  being  each  greater  than  unity,  is  always  real,  so  that  the  ray  can  enter  the  crystal  however  oblique 
its  incidence. 

3x2 


500  LIGHT. 

Light.         Case  2.  When  the  axis  lies  in  the  surface,  orX  =  90°  ;  A  =  a2  ;  B  =  6» ;  C  =  0,  and  the  equations  become       Part  IV- 

7Qc  a  .  sin  0  .  sin  w 

,  •  tan  ff  .  sin  ro  =  — —                                                                                (\R\ 

™-  wllc,n  V  1  -  sin  tf2  {  a- .  sin  ro2  +  6' .  cos  57*  [  ' 
the  axis  lies 

in  the  >ur-  6^                               sin  o  .  cos  w 

fice  tan0.  cosw'  = —  .                                                                  ._;                      (19) 

«       V  1  —  sin  #-  {  o2 .  sin  W  +  i2  .  cos  ro2  } 


«2  /  /.  V 

tan  or  =  -77- .  tan  •ss  =  (  —7-  )  .  tan  w  . 

b*  \  fi'  J 


(20) 


The  latter  of  these  equations  shows  that  the  extraordinary  ray  deviates  from  the  plane  of  incidence.  The  amount 
of  this  deviation  is  nothing  when  the  plane  of  incidence  coincides  with  the  principal  section,  but  increases  on 
either  side  of  it  till  it  attains  a  certain  magnitude,  the  deviation  being  from  the  axis,  or  the  plane  of  refraction 
making  a  greater  angle  with  the  axis  than  that  of  incidence.  The  two  planes  then  approach  each  other,  and 
when  w  =  90°,  tan  OT  =  oc.tan  w'  =  oo,  and,  consequently,  OT'  =  90°,  or  the  plane  of  refraction  coincides  with 
that  of  incidence. 

799.  The  equations  (18)  and  (19)  show  that  in  the  present  case,  the  refracted  ray  does  not  describe  a  conical  surface 
Case  of  about  the  perpendicular  when  the  incident  one  does  so,  and  therefore  that  the  law  of  refraction  varies  in  every 
efraction  in  different  azimuth.  Two  cases  deserve  express  notice,  viz.  those  in  which  the  plane  of  incidence  is  coincident 
pal  sectio'ii  w'1'1  ^e  principal  section,  and  when  perpendicular  to  it.  In  the  former,  w  =  0  and  CT'  =  0,  so  that  we  have 

(21) 


A  remarkable  relation  holds  good  in  this  case  between  the  angles  of  refraction  of  the  ordinary  and  extraordinary 
ray,  their  tangents  being  to  each  other  in  a  given  ratio.  In  fact,  if  we  find  (#')  =  the  angle  of  refraction  for  the 

ordinary  ray,  we  have  sin  (&')  =   —  .  sin  0  =  6  .  sin  0,  and,  consequently, 

b  sin  (0')  6 

tan  &  =r  —  .  —=         —  --  =  —  .tan  (0).  (22) 

a       V  1  -  si»  (°y 
In  the  latter  case,  when  the  plane  of  refraction  is  at  right  angles  to  the  axis,  w  =  OT'  —  90°,  and  we  get 

tan  0'  =  -  a  '  S'"  °      -  ;  sin  ff  =  a  .  sin  0.  (23) 

V  1  -  a2  .  sin  0- 

500.  In  this  case,  therefore,  the  sine  of  incidence  is  in  a  given  ratio  to  that  of  refraction,  and  the  extraordinary 

Case  of  re-  . 

fraction  at    refractiOn  is  performed  according-  to  the  same  law  as  the  ordinary,  only  with  a  different  index,  viz.  u'  or  —  .instead 

right  angles  a 

to  the  prin-  i 

cipalsec-     of  fi,  or  —  .     Hence,  if  we  consider  only  this  particular  case,  the  medium  will  appear  to  have  two  indices  of 

tion.  b 

refraction,  an  ordinary  and  an  extraordinary  one. 

501.  It  was  by  a  careful  examination  of  these  cases,  that  Dr.  Wollaston  was  enabled  to  verify  the  Huygenian  law. 
Experimen-  >phe  circumstance  last  mentioned  puts  it  in  our  power  to  determine  in  the  case  of  any  particular  crystal  the  axes  of 

its  spheroid  of  double  refraction.  We  have  only  to  cut  a  prism  of  it,  having  its  refracting  angle  parallel  to  the 
m.ning  the  ax's>  and  ascertain  its  indices  of  refraction  according  to  the  principles  laid  down  in  the  former  part  of  this  Essay, 
spheroid  of  \  \ 

double  re-    and  calling  them  p.  and  /,  the  semiaxes  of  the  spheroid  will  be  respectively  —  and  —  .     Thus,  in  the  instance 

fraction.  /*  /* 

of  carbonate  of  lime,  which  Malus  examined  with  the  utmost  care,  he  found  the  two  values  of  a  and  b  to  be 
respectively  equal  to  the  numbers  0.67417  and  0.60449,  having  determined  /»'  =  1.4833,  and  p.  =  1.6543. 
(Theorie  de  la  Double  Refraction,  p.  199.) 

802.  In  this  arrangement,  however,  it  is  not  possible  to  decide  simply  from  the  phenomena  of  refraction,  which  is 
the  ordinary,  and  which  the  extraordinary   ray.     There  are,  however,  infallible  and  easy  criteria,  as  we  shall 
speedily  show.     Meanwhile,  we  may  for  the  present  content  ourselves  with  observing,  that  as  a  moderate  devia- 
tion from  the  exact  azimuth  OT  =  90°  imparts  to  the   extraordinary  ray  a  deviation  from  the  plane  of  incidence 
which  does  not  happen  to  the  ordinary  one,  this  may  serve  for  a  criterion  to  distinguish  them  in  certain  cases. 

803.  The  square  of  the  velocity  of  the  ordinary  ray  within  the  medium  is  /i*  VJ,  or  /»2,  that  is.  —  ,  and  is   constant. 
Law  of  the  -,-j  ,0         •     .j.2 

...crementof  That  rf  tfae  extraordinary  •„  V,s>  or   J!_  that  ig  to  say>  ^_  +  « 


or,  V'« 


= | —  ) .  sin  0*. 

i4        \b''         a? ) 


LIGHT.  501 

Light.      The  square  of  the  velocity  of  the  extraordinary  ray  is  therefore  (in  the  corpuscular  doctrine)  diminished  by  a  quantity    Part  IV . 
«->w^-^'  proportional  to  the  square  of  the  sine  of  the  inclination  of  the  ray  within  the  crystal,  to  the  axis.    We  say  dimin-  *>— -v— >s 
ished,  in  the  algebraical  sense  of  the  word,  supposing  a  >  b,  this  agrees  with  common  parlance  ;  but  if  a  <  b,  Division  of 
then  it  will  be  increased.     This  gives  rise  to  the  subdivision  of  the  crystallized  bodies  now   treated  of  into  two  pry818'8 
classes,  which  have  by  some  been  termed  attractive  and  repulsive :  by  others,  positive  and  negative,  which  seems  ™*°  ^'~ 
preferable,  as  the  former  phrases  involve  theoretical  considerations.     Positive  crystals  are,  then,  such. as  have  a  negative. 

less  than  6,  or  in  which  the  spheroid  of  double  refraction  is  prolate.     In  these  the  coefficient  —   (  —  —    — -  \ 

which  we  call  k  is  positive,  and  the  square  of  the  velocity,  or  v*  -f-  A  .  sin  0*,  (where  v  =  —    =  velocity  of  the 

ordinary  ray  within  the  medium,)  is  increased  by  the  action  of  the  medium,  and  is  a  minimum  in  the  axis.  In  the 
negative  class  the  coefficient  k  is  negative,  a  >  b,  or  the  spheroid  of  double  refraction  is  oblate,  and  the  velocity 
of  the  extraordinary  ray  is  a  maximum  along  the  axis.  In  positive  crystals,  therefore,  the  index  of  ordinary 
refraction  O)  is  less  than  that  of  extraordinary  ;  in  negative,  greater.  To  the  former  class  belong  quartz,  ice, 
zircon,  apophyllite,  (when  uniaxal ;)  and  to  the  latter,  Iceland  spar,  tourmaline,  beryl,  emerald,  apatite,  &c. 
The  negative  class,  as  far  as  our  present  knowledge  extends,  far  out-numbers  the  positive  among  natural  and 
artificial  crystals.  They  were  first  distinguished  by  M.  Biot. 

In  the  undulatory  doctrine  the  velocity  is  the  reciprocal  of  what  it  is  in  the  corpuscular  doctrine,  and  is       304. 
therefore  directly  as  the  radius  of  the  spheroid  of  double  refraction.     Hence  a  wave  propagated   within    the  Undulationi 
crystal  from  any  point  will  run  over  in  the  same  time  in  different  directions,  distances  proportional  to  the  radii  propagated 
of  the  spheroid  parallel  to  those  directions ;  and  therefore  at  any  instant  the  surface  of  the  whole  wave  will  be  '"  spte- 
itself  a  spheroid  similar  to  the  spheroid  of  double  refraction.     This  is  Huygens's  conception  of  the  subject.     It  ^'rfoces 
requires  us  to  regard  the  crystal,  or  the  ether  within  the  crystal  through  which  the  undulation  is  propagated,  as 
having  different  elasticities  in  different  directions.     As  far  as  regards  the  molecules  of  a  solid  body  there  is  no 
apparent  impossibility  or  improbability  in  such  an  idea,  but  the  contrary ;  but  if  we  regard  the  propagation  of 
the  light  within  the  medium  to  take  place  by  the  elasticity  of  the  ether  only,  we  must  then  suppose  its  molecules 
in  crystallized  bodies  to  be  in  a  very  different  physical  state  from  what  they  are  in  free  space,  and  either  to  be 
in  some  manner  connected  with   the  solid  particles,  (forming   atmospheres,  for  instance,  about  them,)  or  as 
subjected  to  laws  of  mutual  action  which  approximate  to  those  governing  the  molecules  of  solid  bodies  ;  and 
partaking,  themselves,  of  a  regular  crystalline  arrangement  and  mutual  dependency. 

To  pursue  the  particular  applications  of  the  general  formulae  (13,)  14,)  and  (15)  farther,  would  be  far  beyond       §05 
our  limits.     The  reader  who  is  curious  on  this  very  interesting  part  of  Physical  Optics,  and  who  wishes  to  be  Malns's ' 
delighted  and  instructed  by  a  combination  of  consummate  mathematical  skill  with  sound  experimental  research,  further 
which  may  deservedly  be  cited  as  a  model  of  the  kind,  will  find  every  thing  which  relates  to  the  subject  in  its  researches- 
best  form  in  the  work,  already  so  often  cited,  of  Malus,  Theorie  de  la  Double  Refraction,  which  gained  the  mathe- 
matical prize  of  the  French  Institute  in  1810.     To  the  theory  of  the  internal  reflexion  of  the  extraordinary  ray 
which  offers  many  remarkable  particularities,   as  there  delivered,  we  must  especially  refer  him,  as  well  as  to 
his  investigation  of  the  foci  of  lenses  formed  of  doubly  refracting  crystals,  of  which  we  shall  here  only  extract  Foci  of  a 
the  results,  in  the  single  case  of  a  double  convex  lens  having  the  axis  of  double  refraction  in  the  direction  doubly 
of  the  axis  of  the  lens.  refracting 

Let  r,  r1  be  the  radii  of  the  anterior  and  posterior  surfaces  of  the  lens,  both  supposed  convex. 

d  =  distance  of  the  radiant  point  in  the  axis. 

a,  b  =  the  equatorial  and  polar  radii  of  the  spheroid  of  double  refraction,  as  above. 
D  =  distance  of  the  conjugate  focus  behind  the  lens  for  extraordinary  rays. 
A  =  extraordinary  focal  length  for  parallel  rays. 
F  :=  ordinary  focal  length  for  parallel  rays. 
Then  shall  we  have  for  the  general  expression  of  D, 

a^bdrr1 _         -brr1 


(r-f  r1)  (1  -6)  ' 

If  the  lens  be  equi-convex,  or  r  =  r1, 

_  a*b  r  d  __  a?br 


•2  (2  b*  -  a*  -  a*  6)  d  -  aa  6  r  '  2  (2  6*  -  a*  -  a*  6)  ' 


F—   « 


2(1  -b)'  '  2  62  -  a«  -  a*  b  ' 

In  the  case  of  Iceland  spar,  these  last  equations  become 

D  =  - r  .  88,2286 ;        F  =-  r  .  0,7642  ;         D  -  F  =-  F  .  114,4546  ; 
and  in  the  case  of  rock  crystal  (quartz) 

D  =  -  r  .  0.9628  ;        F  =  -  r  .  0,8958 ;         D  -  F  =  -  F.0,0748. 

To  represent,  in  general,  the  course  of  any  extraordinarily  refracted  ray,  Huygens  has  giving   the  following       806. 
construction,  (fig.  170.)     Let  H  E  D  be  the  elliptic  section  of  the  spheroid  of  double  refraction  by  the  surface, 
and  RC  the  incident  ray  falling  on  C  its  centre,  and  B  C  K  the  orthographic  projection  of  the  ray  R  C  on  the 


502  LIGHT. 

i-ight.  surface.  Let  H  M  E  be  the  portion  of  the  spheroid  within  the  crystal,  whose  axis  passes  through  C,  and  may  be 
>— -\— -'  anyhow  inclined  to  the  surface.  Then  will  the  surface  of  this  spheroid  be  the  boundary  of  the  wave  propagated  v 
Huygens's  from  C  as  a  centre,  after  the  lapse  of  a  given  time.  Draw  CO  in  the  plane  R  C  K  at  right  angles  to  R  C,  and 
Son'for'ex-  ma'ie  ^  ^  (perpendicular  to  C  K,  or  parallel  to  R  C)  equal  to  the  space  described  by  light  in  the  medium 
iraordiauT*  exterior  to  the  crystal  in  the  same  given  time.  This  will  determine  the  point  K  in  the  line  B  C  K.  Through  K 
refraction."  draw  K  T  perpendicular  to  B  K,  and  about  K  T  as  an  axis  let  a  plane  revolve  passing  through  K  T,  till  it  touches 
Fig.  170.  the  surface  of  the  spheroid  in  I.  Join  C  I,  and  C  I  is  the  extraordinary  refracted  ray. 

807.  The  demonstration  of  this  construction  (granting  the  principle  of  spheroidal  undulations)  is  evident,  if  we 
Demonstra-  consider  the  manner  in  which  the  general  wave,  a  perpendicular  to  whose  surface  forms  what  we  term  a  ray  of 
tion  from     light,  (at  least  in  singly  refracting  media,)  arises  from  the  reunion  of  all  the  elementary  waves  propagated  from 

e  of  sThe  everv  Par'  °^  *ne  surface.  (Art.  586.)     In  this  construction,  if  we  conceive  a  plane  wave  from  an  infinitely  distant 

Toidal  on-    luminary  perpendicular  to  R  C  to  move  along  R  C,  every  point  in  the  line  C  K  will  become  in  succession,  and 

Uulations.     every  point  in  the  line  C  D  perpendicular  to  C  K,  or  parallel  to  KT  simultaneously,  a  centre  of  vibration.     The 

general  wave,  therefore,  will  be  a  surface  touching  all  ellipsoids  described  about  each  point  of  the  surface,  having 

their  axes  parallel,  their  generating  ellipses  similar,  and  their  linear  dimensions  proportional  to  the  distance  of 

their  centre  from  the  line  KT.     Of  course  it  can  be  no  other  than  the  tangent  plane  I  KT  drawn  as  above. 

808.  This  then  will  be  the  form  and  position  of  the  general  wave  within  the  crystal.     Now  if  we  consider  only  that 
very  minute  portion  of  it  which  emanates  from  C,  it  is  evident  that  I  is  the  corresponding  point  in  it;    and 
therefore  C  I  is  necessarily  the  direction  of  the  ray,  because  I  is  the  point  on  which  that  portion  of  the  general 
wave  transmitted  through  a  very  small  aperture  at  C  would  fall. 

809.  Thus  we  see,  that  in  the  case  of  the  extraordinary  ray,  we  are  no  longer  to  regard  the  ray  as  a  perpendicular 
Oblique       to  the  surface  of  the  wave.     It  is  propagated  obliquely  to  that  surface.     So  soon,  however,  as  the  wave  emerges 
propagation  jnjo  jne  ambient  medium,  the  usual  law  of  perpendicular  propagation  is  restored. 

sxtra-      r£Q  snow  tjje  identity  of  the  law  of  extraordinary  refraction  resulting  from  this  construction  with  that  expressed 

810       "y  *ne  genera'  equations  (13,)  (14,)  and  (15,)  we  have  only  to  translate  it  into  analytical  language.     This  has 

been  done  by  Malus,  in  his  work  above  referred  to ;  and  the  reader  may  also  consult  Biot's  Traite  General  de 

Physique,  for  a  more  elementary  exposition  of  the  process,  which  is  one  of  considerable  complexity,  for  which 

reason  we  shall  not  embarrass  ourselves  with  it  here. 

SH.  Some  very  remarkable  and  important  consequences  follow  front  this  mode  of  viewing  the  subject.     It  appears 

Form  and     that  when  a  plane  wave  is  incident  on  a  doubly  refracting  surface,  the  transmitted  extraordinary  wave  is  also 

position  of  plane,  and  advances  with  a  uniform  velocity  in  a  direction  oblique  to  itself.     Consequently  the  velocity  is  also 

the  extraor-  unjform  jn  a  direction  perpendicular  to  itself.    Moreover,  its  common  section  with  the  surface  is  always  parallel  to 

dmary  ray.  ^  r^  Qr  ^o  ^  comlnon  section  of  the  incident  wave  with  the  same  surface.     Hence,  it  is  evident,  that  it  moves 

in  the  same  way  as  an  ordinarily  transmitted  wave  would  do,  and  at  any  instant  has  the  same  position  that  such 

a  wave  would  have,  provided  the  index  of  refraction  in   the   latter  case  were  properly   assumed.     The   only 

difference  is,  that  the  motions  of  the  vibrating  molecules,  of  which   they  respectively  consist,  are  executed  in 

different  planes.     Now,  when  this  wave  emerges  from  the  medium,  it  obeys  the  same  laws  as  on  its  entry,  only 

reversed  ;  so  that  it  still  continues  a  plane  wave,  and  its  common  section  with  the  surface  of  emergence  remains 

unaltered. 

812.  Hence  it  follows,  that  if  we  cut  a  prism  of  any  doubly  refracting  crystal  with  one  axis,  and  transmit  through 
Oonse-        it  a  ray  incident  in  a  plane  at  right  angles  to  the  edge  of  the  prism,  the  ordinary  and  extraordinary  ray  will  both 
quences  in    emerge  in  that  plane,  and  their  separation  will  take  place  in  a  plane  containing  the  incident  and  ordinarily- 
refracted  ray,  and  will  therefore  be,  apparently,  such  as  would  arise  from  attributing  two  ordinary  refractive 

through       powers  to  the  medium.     It  is  only  when  the  edge  of  the  prism  is  oblique  to  the  plane  of  incidence,  that  the 
prisms.         extraordinary  ray  can  deviate  from  the  plane  containing  the  incident  and  ordinarily  refracted  rays. 

813.  We  see,  then,  that  in  the  theory  of  extraordinary  refraction,  it  is  necessary  to  consider,  as  distinct,  two  things, 
Velocity  of  which,  in  that  of  ordinary,  are  one  and  the  same,  viz.  the  velocity  of  the  luminous  waves,  and  the  velocity  of  the 
lummoui      ray!l  Oj-  light.     This  distinction  will  require  to  be  very  carefully  kept  in  view  hereafter,  when  we  come  to  treat 
of"',^'  o/    of  the  law  of  refraction  in  crystals  with  two  axes  of  double  refraction.     For  this,  however,  we   are  not  yet 
lijht  dis-     prepared,  as  the  knowledge  of  this  law  presupposes  an  acquaintance  with  a  multitude  of  facts  relative  to  the 
tmguished.  polarization  of  light,  of  which  we  have  yet  said  nothing.     It  will  suffice  here  to  mention,  that  the  whole  doctrine 
Theory  of    of  double  refraction  has  recently  undergone  a  great  revolution  ;  one,  indeed,  which  may  be  said  to  have  changed 
double  re-    the  face  of  Physical  Optics,  in  consequence  of  the  researches  of  M.  Fresnel.     It  had  all  along  been  taken  for 
fraction  in    granted,  that  in  crystals  possessed  of  double  refraction,  one  of  the  pencils  followed  the  ordinary  law  of  propor- 
tional sines.     It  had,  moreover,  been  ascertained,  by  experiments  hereafter  to  be  related,  that  the  difference  of 

deferred,      the  squares  of  the  velocities  between  the  two  pencils  is  in  all  cases  proportional  to  the  product  of  the  sines  of 

and  why      tne  anff'es  contained  between  the  extraordinary  ray  (as  it  was  termed)  and  the  two  axes,  or  directions  in  which 

the  refraction  is  single.     It  wag  hence  concluded,  that  the  velocity  of  the  extraordinary  pencil  was  in  all  cases 

represented  by  ^  v*  -\-  k  .  sin  (j> .  sin  0',  v  being  that  of  the  ordinary  one,  and  k  a  constant  depending  on  the 
nature  of  the  crystal,  and  0,  0'  the  angles  in  question.  This  granted,  there  would  be  no  difficulty  in  deter- 
mining the  form'of  the  surface  of  double  curvature,  which  should  be  substituted  for  the  Huygenian  spheroid; 
so  as  to  render  the  same  construction  with  that  described  in  Art.  806,  or  the  general  formula  in  Art.  792,  appli- 
cable to  this  case.  In  fact,  if  we  call  a  the  semi-angle  between  the  two  axes,  and  conceive  three  coordinates  x, 
y,  z,  of  which  x  bisects  that  angle,  the  plane  of  the  x,  y  containing  both  axes,  it  is  easy  to  see,  by  spherical 
trigonometry,  that  we  must  have 


L  I  G  H  T.  503 

.  __   x  .  cos  a  -f-  y  .  sin  a  .. x  .  cos  a  —  y  .  sin  a  Part  IV. 

V  x'1 -+- y- +  z*  V  J*  -f-  y«  -j-  s»  y-— v~- 


Hence,  since  r  (^  x*  -f-  ye  -j-  2*)  the  radius  of  the  surface  of  the  wave,  is  always  equal  to 


1  1 

or 


V  • 

</&+!?.  sin  0  .  sin 


a  simple  substitution  would  give  at  once  the  equation  of  its  surface  as  referred  to  the  three  coordinates  4',  y,  z  ; 
namely, 

0  =  (if  -  r>4)  (xl  +  y1  +  z8)"  +  2  (**  +  y*  -f-  z2)  (»«  -  fc8  a?  .  cos  a8  -  If  y*  .  sin  a8) 

-f-  &"  (x2  .  cos  a*  -f  y!  .  sin  a»)»  —  1, 

which  it  would  be  easy  then  to  transform  into  functions  of  r,  is,  and  0,  as  required  for  the  application  of  the 
general  analytical  formula?  by  the  usual  substitutions 

z  =  r  .  sin  0  ;        y  =  r  .  sin  0  .  sin  zs  ;        x  =:  r  .  sin  0  .  cos  •a. 

The  researches  of  M.  Fresnel,  however,  as  before  remarked,  have  destroyed  the  basis  on  which  this  theory 
rested,  by  demonstrating  the  non-existence  of  an  ordinarily  refracted  ray  in  the  case  of  crystals  with  two  axes. 
The  theory  which  he  has  substituted  in  its  place,  however,  and  which  it  is  impossible  to  regard  otherwise  than 
as  one  of  the  finest  generalizations  of  modern  science,  we  must  reserve  for  a  more  advanced  place  in  this  essay. 
We  shall  now  proceed  to  treat 

Of  the  Polarization  of  Light, 

The  phenomena  which  belong  to  this  division  of  our  subject  are  so  singular  and  various,  that  to  one  who  has  §14. 
only  studied  the  subject  of  Physical  Optics  under  the  relations  presented  in  the  foregoing  pages,  it  is  like  enter- 
ing into  a  new  world,  —  so  splendid  as  to  render  it  one  of  the  most  delightful  branches  of  experimental  inquiry  ; 
and  so  fertile  in  the  views  it  lays  open  of  the  constitution  of  natural  bodies,  and  the  minuter  mechanism  of  the 
universe,  as  to  place  it  in  the  very  first  rank  of  the  physico-mathematical  sciences,  which  it  maintains,  by  the 
rigorous  application  of  geometrical  reasoning  its  nature  admits  and  requires.  The  intricacy  as  well  as  variety 
of  its  phenomena,  and  the  unexampled  rapidity  with  which  discoveries  have  succeeded  each  other  in  it,  have 
hitherto  prevented  the  possibility  of  embodying  it  satisfactorily  in  a  systematic  form  ;  but,  after  the  rejection  of 
numberless  imperfect  generalizations,  it  seems  at  length  to  have  acquired  that  degree  of  consistency  as  to  enable 
us  —  not,  indeed,  to  deduce  every  phenomenon,  by  distinct  steps,  from  one  general  cause  —  but  to  present  them, 
at  least,  in  something  like  a  regular  succession  ;  to  show  a  mutual  dependence  between  their  several  classes, 
which  is  a  main  step  to  a  complete  generalization  ;  and  to  dispense  with  the  bewildering  detail  of  an  immense 
multitude  of  individual  facts,  which,  having  served  their  purpose  in  the  inductive  process,  must  in  future  be 
considered  as  having  their  interest  merged  in  that  of  the  laws  from  which  they  flow. 


§  II.  General  Ideas  of  the  Distinction  between  Polarized  and  Unpolarized  Light. 

In  all  the  properties  and  affections  of  light  which  we  have  hitherto  considered,  we  have  regarded  it  as  3 [5 
presenting  the  same  phenomena  of  reflexion  and  transmission,  both  as  respects  the  direction  and  intensity  of 
the  reflected  or  transmitted  beam,  however  it  may  be  presented  to  the  reflecting  or  refracting  surface,  provided 
the  angle  of  incidence,  and  the  plane  in  which  it  lies,  be  not  varied.  And  this  is  true  of  light  in  the  state  in 
which  it  is  emitted  immediately  from  the  sun,  or  from  other  self-luminous  sources.  A  ray  of  such  light,  incident 
at  a  given  angle  on  a  given  surface,  may  be  conceived  to  revolve  round  an  axis  coincident  with  its  own  direction  ; 
or,  which  comes  to  the  same  thing,  the  reflecting  or  refracting  surface  may  be  actually  made  to  revolve  round  the 
ray  as  an  axis,  preserving  the  same  relative  situation  to  it  in  all  other  respects,  and  no  change  in  the  phenomena 
will  be  perceived.  For  instance,  if  in  a  long  cylindrical  tube  we  fix  a  plate  of  glass,  or  any  other  medium  at 
any  angle  of  inclination  to  the  axis ;  and  then,  directing  the  tube  to  the  sun,  turn  the  whole  apparatus  round  on 
its  axis,  the  intensity  of  the  reflected  or  refracted  ray  will  suffer  no  variation,  and  its  direction  (if  deviated)  will 
revolve  uniformly  round  with  the  apparatus,  so  that  if  received  on  a  screen  connected  invariably  with  the  tube,  it 
will  continue  to  fall  on  the  very  same  point  in  all  parts  of  its  rotation.  Or  we  may  receive  the  light  from  a 
piece  of  white  hot  iron  at  any  angle  on  any  medium,  and  its  phenomena  will  be  precisely  the  same,  whether  the 
iron  be  at  rest,  or  be  made  to  revolve  round  an  axis  coincident  with  the  direction  of  the  ray. 

But,  if  instead  of  employing  a  ray  immediately  emitted  from  a  self-luminous  source,  we  subject  to  the  same  gig. 
examination  a  ray  that  has  undergone  some  reflexions,  refractions,  or  been  in  any  one  of  a  great  variety  of  Polarized 
ways  subjected  to  the  action  of  material  bodies,  we  find  this  perfect  uniformity  of  result  no  longer  to  hold  good.  rays  have 
It  is  no  longer  indifferent  in  what  plane,  with  respect  to  the  ray  itself,  the  reflecting  or  refracting  surface  is  »C(lulred 
presented  to  it.  It  seems  to  have  acquired  sides ;  a  right  and  left,  a  front  and  back  ;  and  the  intensity,  though  ti*^s  ™  ' 
not  the  direction  of  the  reflected  or  transmitted  portion,  depends  materially  on  the  position  with  respect  to  these  external 

space. 


504  LIGHT. 

Light,  sides,  in  which  the  plane  of  incidence  lies,  though  everything  else  remains  precisely  the  same.  In  this  state  it  is 
— -v-^— •  said  to  be  polarized.  The  difference  between  a  polarized  and  an  ordinary  ray  of  light  can  hardly  be  more  readily 
Illustration,  conceived  than  by  assimilating  the  latter  to  a  cylindrical,  and  the  former  to  a  four-sided  prismatic  rod,  such  as  a 
lath  or  a  ruler,  or  other  long,  flat,  straight  stick.  It  is  evident  that  the  cylinder,  if  inclined  to  any  surface  at  a 
given  angle  in  a  given  plane,  may  be  turned  round  its  own  axis  without  altering  its  relations  to  the  plane,  while 
those  of  the  prism  will  vary  essentially  according  to  the  position  of  its  sides.  Let  us  suppose,  for  instance,  (it 
is  but  a  simile,  which  we  do  not  wish  the  reader  to  dwell  on  for  a  moment,  or  to  imagine  that  any  analogy  is 
hereafter  intended  to  be  established,)  that  we  had  occasion  to  thrust  such  a  rod  into  a  surface  composed  of 
detached  fibres,  all  lying  in  one  direction,  or  of  scales  or  laminae  arranged  parallel  to  one  another,  we  should 
.find  a  much  greater  facility  of  penetration  on  presenting  its  broad  side  in  the  direction  of  the  laminae  or  fibres, 
than  transverse  to  them.  A  thin  sheet  may  be  slipped  between  the  bars  of  a  grating,  which  would  present  an 
insuperable  obstacle  to  it  if  presented  cross-wise. 

817.  But,  to  be  more  particular,  and  to  give  a  more  clear  conception  of  the  marked  distinction  which  exists  between 
Property  of  a  polarized  and  an  unpolarized  ray.     There  are  many  crystallized  minerals,  which  when  cut  into  parallel  plates 

tourma-  are  sufficiently  transparent,  and  let  pass  abundance  of  light  with  perfect  regularity,  but  which,  nevertheless,  at 
o'tta^crvs-  *ts  emer£ence  's  found  to  have  acquired  that  peculiar  modification  here  in  question.  One  of  the  most  remark- 
tab,  able  of  these  is  the  tourmaline.  This  mineral  crystallizes  in  long  prisms,  whose  primitive  form  is  the  obtuse 
rhomboid,  having  its  axis  parallel  to  the  axis  of  the  prism.  The  lateral  faces  of  these  prisms  are  frequently  so 
numerous  as  to  give  them  an  approach  to  a  cylindrical  or  cylindroidal  form.  Now  if  we  take  one  of  these 
crystals,  and  slit  it  (by  the  aid  of  a  lapidary's  wheel)  into  plates  parallel  to  the  axis  of  the  prism  of  moderate 
and  uniform  thickness,  (about  -£$  of  an  inch,)  which  must  be  well-polished,  luminous  objects  may  be  seen 
through  them,  as  through  plates  of  coloured  glass.  Let  one  of  these  plates  be  interposed  perpendicularly 
between  the  eye  and  a  candle,  the  latter  will  be  seen  with  equal  distinctness  in  every  position  of  the  axis  of  the 
plate  with  respect  to  the  horizon,  (by  the  axis  of  the  plate  is  meant  any  line  in  it  parallel  to  the  axes  of  its 
molecules,  or  to  the  axis  of  the  prism  from  which  it  was  cut.)  And  if  the  plate  be  turned  round  on  its  own 
plane,  no  change  will  be  perceived  in  the  image  of  the  candle.  Now,  holding  this  first  plate  in  a  fixed  position, 
(with  its  axis  vertical,  for  instance,)  let  a  second  be  interposed  between  it  and  the  eye,  and  turned  round  slowly  in 
its  own  plane,  and  a  very  remarkable  phenomenon  will  be  seen.  The  candle  will  appear  and  disappear  alternately 
at  every  quarter  revolution  of  the  plate,  passing  through  all  gradations  of  brightness,  from  a  maximum  down 
to  a  total,  or  almost  total,  evanescence,  and  then  increasing  again  by  the  same  degrees  as  it  diminished  before. 
If  now  we  attend  to  the  position  of  the  second  plate  with  respect  to  the  first,  we  shall  find  that  the  maxima  of 
illumination  take  place  when  the  axis  of  the  second  plate  is  parallel  to  that  of  the  first,  so  that  the  two  plates 
have  either  the  same  positions  with  respect  to  each  other  that  they  had  in  the  original  crystal,  or  positions  differing 
by  180°,  while  the  minima,  or  evanescences  of  the  image,  take  place  exactly  90°  from  this  parallelism,  or  when 
the  axes  of  the  two  plates  are  exactly  crossed.  In  tourmalines  of  a  good  colour,  the  stoppage  of  the  light  in 
this  situation  is  total ,  and  the  combined  plate  (though  composed  of  elements  separately  very  transparent  and  of 
the  same  colour)  is  perfectly  opake.  In  others  it  is  only  partial ;  but  however  the  specimens  be  chosen,  a  very 
marked  defalcation  of  light  in  the  crossed  position  takes  place.  We  shall  at  present  suppose  that  the  specimens 
employed  possess  the  property  in  question  in  its  greatest  perfection.  Now  it  is  evident  that  the  light  which  has 
passed  through  the  first  plate  has  acquired  in  so  doing  a  property  totally  distinct  from  those  of  the  original  light 
of  the  candle.  The  latter  would  have  penetrated  the  second  plate  equally  well  in  all  its  positions  ;  the  former  is 
incapable  altogether  of  penetrating  it  in  some  positions,  while  in  others  it  passes  through  readily,  and  these 
positions  correspond  to  certain  sides  which  the  ray  has  acquired,  and  which  are  parallel  and  perpendicular 
respectively  to  the  axis  of  the  first  plate.  Moreover,  these  sides  once  acquired,  are  retained  by  the  ray  in  all  its 
future  course,  (provided  it  be  not  again  otherwise  modified  by  contact  with  other  bodies,)  for  it  matters  not  how 
great  the  distance  between  the  two  plates,  whether  they  be  in  contact  or  many  inches,  yards,  or  miles  asunder, 
not  the  least  variation  is  perceived  in  the  phenomenon  in  question.  If  the  position  of  the  first  plate  be  shifted, 
the  sides  of  the  transmitted  ray  shift  with  it,  through  an  equal  angle,  and  the  second  will  no  longer  extinguish 
it  in  the  position  it  at  first  did,  but  must  be  brought  into  a  position  removed  therefrom,  by  an  angle  equal  to 
that  through  which  the  first  plate  has  been  made  to  revolve. 

818.  A  great  many  other  crystallized  bodies  besides  the  tourmaline  possess  this  curious  property,  and  several  in 
Selection  of  great  perfection.     The  tourmaline,  however,  is  one  easily  procured,  and  being  exceedingly  useful  in  optical 

experiments,  we  would  recommend  the  reader  who  has  any  desire  to  familiarize  himself  with  the  practical 
"es-  manipulations  of  this  branch  of  optical  science,  to  provide  himself  with  a  good  pair  of  corresponding  plates  of 
this  mineral,  cut  and  polished  as  above  directed.  The  colour  is  a  point  of  great .  moment.  Those  of  a  blue 
or  green  colour  possess  the  property  in  question  very  imperfectly  ;  the  yellow  varieties,  unless  when  verging  to 
greenish  brown,  are  equally  improper,  the  best  colour  is  a  hair-brown,  or  purplish  brown,  and  they  may  be  slit 
and  polished  by  any  lapidary. 

819.  But  it  is  not  only  by  such  means  that  the  polarization  of  a  pencil  of  light  may  be  operated,  nor  is  this  the  only 
Various        character  which  distinguishes  polarized  from  ordinary  light.     We  shall,  therefore,  describe  in  order,  the  principal 
modes  of      means  by  which  the  polarization  of  light  may  be  performed,  and  the  assemblage  of  characters  which  are  inva- 
rmiarmng     rjably  found  to  coexist  in  a  ray  when  polarized. 

The  chief  modes  by  which  the  polarization  of  lig'ht  may  be  effected,  are 
1st.  By  reflexion  at  a  proper  angle  from  the  surfaces  of  transparent  media. 

2d.  By   transmission   through   a  regularly  crystallized   medium   possessed   of  the   property   of  double  re- 
fraction. 

3d.  By  transmission  through  transparent,  uncrystallized  plates  in  sufficient  number,  and  at  proper  angles. 


LIGHT.  505 

Light.          4th.  By  transmission  through  a  variety  of  bodies,  such  as  agate,  mother-of-pearl,  &c.  which  have  an  approach     ^art  Iv- 
"v*r/  to  a  laminated  structure,  and  an  imperfect  state  of  crystallization.  v-— v--— 

The  characters  which  are  invariably  found  to  coexist  in  a  polarized  ray,  being  the  chief  of  those  by  which  it       820. 
may  be  most  easily  recognised  as  polarized,  are —  Characters 

1.  Incapability  of  being  transmitted  by  a  plate  of  tourmaline,  as  above  described,  when  incident  perpendicu-  °.f  a  Pola" 
larly  on  it,  in  certain  positions  of  the  plate  ;   and  ready  transmission  in  others,  at  right  angles  to  the  former.        "niokt* 

2.  Incapability  of  being  reflected  by  polished  transparent  media  at  certain  angles  of  incidence,  and  in  certain 
positions  of  the  plane  of  incidence. 

3.  Incapabiltiy  of  undergoing  division  into  two  equal  pencils  by  double  refraction,  in  positions  of  the  doubly 
refracting  bodies,  in  which  a  ray  of  ordinary  light  would  be  so  divided. 

Besides  which,  there  might  be  enumerated  a  vast  variety  of  other  characters,  which,  however,  it  will  be  better 
to  regard  as  properties  at  once  of  polarized  light,  and  of  the  various  media  which  affect  it.     It  cannot  fail  to  be 
remarked,  that  all    these  characters  are  of  the  negative  kind,  and  consist  in  denying  to  polarized  light  properties 
which  ordinary  light  possesses,  and  that  they  are  such  as  affect  the  intensity  of  the  ray,  not  its  direction.     Thus,  Affect  the 
the  direction  which  a  polarized  ray  will  take  tinder  any  circumstances  of  the  action  of  media,  is  never  different  intensity 
from  what  an  unpolarized  ray  might  take,  and  from  what  a  portion  of  it  at  least  actually  does.     For  instance,  and  "?' tlle 
•     when  an  unpolarized  ray  is  separated  by  double  refraction  into  two  equal  pencils,  a  polarized  ray  will  be  divided  ^"rav'"  °' 
into  two  unequal  ones,  one  of  which  may  even  be  altogether  evanescent,  but  their  directions  are  precisely  the 
same  as  those  of  the  pencils  into  which   the  unpolarized   ray  is  divided.     Hence  we  may  lay  it  down   as  a 
general  principle,  that  the  direction  taken  by  a  polarized  ray,  or  by  the  parts  into  which  it  may  be  divided  by 
any  reflexions,  refractions,  or  other  modifying  causes,  may  always  be  determined  by  the  same  rules  as  apply  to 
unpolarized  light ;   but  that  the  relative  intensities  of  these  portions  differ  from  those  of  similar  portions  of 
unpolarized  light,  according  to  certain  laws  which  it  is  the  business  of  the  optical  inquirer  to  ascertain. 


§  III.     Of  the  Polarization  of  Light  by  Reflexion. 

When  a  ray  of  direct  solar  light  is  received  on  a  plate  of  polished  glass  or  other  medium,  a  portion  more  or       821. 
less  considerable  is  always  reflected.     The   intensity  of  this  portion  depends  only  on  the  nature  of  the  medium  Light 
and  on  the  angle  of  incidence,  being  greater  as  the  refractive  power  of  the  former  is  greater,  and  as  the  ray  falls  Pola"?e(1  by 
more  obliquely  on  the  surface.  But  it  is,  moreover,  found,  that  at  a  certain  angle  of  incidence,  (which  is  therefore  Ie' 
called   the  polarizing  angle,)  the  reflected  ray  possesses  all  the  characters  above  enumerated,  and  is  therefore 
polarized. 

This  remarkable  fact  was  discovered  by  Malus  in  1808,  when  accidently  viewing,  through  a  doubly  refracting      822. 
prism,  the  light  of  the  setting  sun  reflected  from  the  glass  windows  of  the  Luxembourg  Palace  in  Paris.     On  Discovery 
turning  round  the  prism,  he  was  surprised  to  observe  a  remarkable  difference  in  the  intensity  of  the  two  images;  by  Malus- 
the  nost  refracted  alternately  surpassing  and  falling  short  of  the   least  in  brightness,   at  each  quadrant  of  the 
revolution.     This  phenomenon  connecting  itself  in  his  mind  with  similar  phenomena  produced  by  rays  which  had 
undergone  double  refraction,  and  with  which,  from  the  researches  he  was  then  engaged  in,  he  was  familiar,  led 
him  to  investigate  the  circumstances  of  the  case  with  all  possible  attention,  and  the  result  was  the  creation  of  a 
new  department  of  Physical  Optics.     So  true  it  is,  that  a  thousand  indications  pass  daily  before  our  eyes  which 
might  lead  to  the  most  important  conclusions.     The  seeds   of  great   discoveries  are  everywhere  present  and 
floating  around  us,  but  they  fall  in  vain  on  the  unprepared  mind,  and  germinate  only  where  previous  inquiry  has 
elaborated  the  soil  for  their  reception,  and  awakened  the  attention  to  a  perception  of  their  value. 

To  make  this  new  property  acquired  by  the  reflected  ray  evident  by  experiment,  let  any  one  lay  down  a  large  823. 
plate  of  glass  on  a  black  cloth,  on  a  table  before  an  open  window,  and  placing  himself  conveniently  so  as  to  look  Experiment 
obliquely  at  it,  l«t  him  view  the  reflected  light  of  the  sky,  (or,  which  is  better,  of  the  clouds  if  not  too  dark,) 
from  the  whole  surface,  which  will  thus  appear  pretty  uniformly  bright.  Then  let  him  close  one  eye,  and  apply 
before  the  other  a  plate  of  tourmaline,  cut  as  above  directed,  so  as  to  have  its  axis  in  a  vertical  plane.  He  will 
then  observe  the  surface  of  the  glass,  instead  of  being  as  before  equally  illuminated,  to  have  on  it,  as  it  were, 
an  obscure  cloud,  or  a  large  blot,  the  middle  of  which  is  totally  dark.  If  this  be  not  seen  at  first,  it  will  come 
into  view  on  elevating  or  depressing  the  eye.  If  the  inclination  of  a  line  drawn  from  the  centre  of  the  dark 
spot  to  the  eye  be  measured,  it  will  be  found  to  make  an  angle  of  about  33°  with  the  surface  of  the  glass.  If 
now,  keeping  the  eye  fixed  on  the  spot,  the  tourmaline  plate  (which  it  is  convenient  to  have  set  in  a  small 
circular  frame  for  such  experiments)  be  turned  slowly  round  in  its  own  plane,  the  spot  will  grow  less  and  less 
obscure,  and  when  the  axis  of  the  tourmaline  is  parallel  to  the  reflecting  surface,  (or  horizontal,)  will  have  dis- 
appeared completely,  so  as  to  leave  the  surface  equally  illuminated,  and,  on  continuing  the  rotation  of  the  tourma- 
line, will  appear  and  vanish  alternately. 

It  appears  from  this  experiment,  that,  the  ray  which   has  been  reflected  from  the  surface  of  the  glass  at  an       824 
inclination  of  33°,  or  an  incidence  of  57°,  has  thereby  been   deprived  of  its  power  to  penetrate  a  tourmaline 
plate  whose  axis  lies  in  the  plane  of  incidence.     It  has  therefore  acquired  the  sam«  character,  or  (so  far  as  this 
goes,  at  least)  undergone  (he  same  modification  as  if,  instead  of  being  reflected  on  glass,  it  had  been  transmitted 
through  a  tourmaline  plate,  whose  axis  was  perpendicular  to  the  plane  of  reflexion. 

It  has,  moreover,  acquired  all  the  other  enumerated  characters  of  a  polarized  ray.     And,  first,  it  has  become 

VOL.  iv.  3  u 


506  L  I  G  H  T. 

Light.  incapable  of  reflexion  at  the  surface  of  glass,  or  other  transparent  media  at  certain   definite  angles,   and  in     Part 

^ -"•"•'  certain  positions  of  the  plane  of  incidence.     To  show  this  experimentally,  let  a  piece  of  polished  glass  have  one  V~^" 

825.  Of  jts   surfaoes  roughened,  and  blackened  with   melted  pitch  or  black  varnish,   so  as  to  destroy  its  internal 
e1"  reflex'on>  and  let  this  De  fixed  on  a  stand,  so  as  to  be  capable  of  varying  at  will  the  inclination  of  its  polished 

rizedPray  surface  to  the  horizon,  and  of  turning  it  round  a  vertical  axis  in  any  azimuth.  A  very  convenient  stand  of  this 
incapable  of  kind  is  figured  in  fig.  171,  consisting  of  a  cylindrical  support  A  sliding  in  a  vertical  tube  B,  attached  to  a  round 
a  second  re-  base  F  like  a  candlestick,  and  carrying  an  arm  C,  which  can  be  set  to  any  angle  of  inclination  to  the  horizon  by 
Bet  ion,  Ac.  m(>ans  of  a  stiff  shoulder  joint  D.  To  this  arm  the  blackened  glass  E  is  fixed,  having  its  plane  parallel  to  the 
axis  of  the  joint  D.  Let  this  apparatus  be  set  on  a  table,  so  that  the  rays  reflected  from  a  pretty  large  plate  of 
glass  G,  at  an  angle  of  about  57°  (of  incidence)  shall  be  received  on  the  glass  E,  which  ought  to  be  inclined 
with  its  polished  surface  looking  downwards,  and  making  an  angle  of  about  73°  with  the  horizon,  see  Art.  842. 
Then  let  the  observer  apply  his  eye  near  the  glass  E,  so  as  to  see  the  glass  G  reflected  in  it,  and  slowly  turn 
the  stand  F  round  in  a  horizontal  plane,  keeping  always  the  reflected  image  of  G  in  view.  He  will  then 
perceive,  that  at  a  certain  point  of  the  rotation  of  the  stand,  the  illumination  of  this  image,  which  in  other 
situations  is  very  bright,  will  undergo  a  rapid  diminution,  and  at  last  wholly  disappear,  and  (if  the  glass  G  be 
large  enough)  the  same  appearance  of  a  cloud  or  large  dark  spot  will  then  be  visible  upon  it.  If  the  inclination 
of  the  arm  C  D  be  correct,  it  will  be  easy  to  find  such  a  position  by  'turning  the  stand  a  little  backwards  and 
forwards,  as  shall  make  the  centre  of  this  spot  totally  black  ;  if  not,  bring  it  to  as  great  a  degree  of  obscurity  as 
possible  by  the  horizontal  motion,  then,  holding  fast  the  stand,  vary  a  little  one  way  or  another  the  inclination  of 
the  reflector  E,  and  a  very  complete  obscurity  will  readily  be  attained. 

826.  Another,  and,  for  some  experimental  purposes,  a  better  way  of  exhibiting  the  same  phenomenon,  is  to  take 
Another        two  metallic  or  pasteboard  tubes,  open  at  both  ends,  and  fitting  into  each  other  so  as  to  turn  stiffly.     Into  each 
mode  of        of  these,  at  the  end  remote   from  their  junction,  fix  with  wax,  or  in  a  frame,  a  plate  of  glass,  blackened  at  the 
making  the    j^^  as  aDOVe  described,  so  as  to  make  an  angle  of  33°  with  the   axis  of  the  tube,  as  represented  in  fig.  172. 
Fi^  172        Then  having  placed  the  tube  containing  one  of  the  plates  (A)  so  that  the  light  from  any  luminary,  reflected  at 

the  plate  shall  traverse  the  axis  of  the  tube,  fix  it  there,  and  the  reflected  ray  will  be  again  reflected  at  B,  and 
on  its  emergence  may  be  received  on  a  screen  or  on  the  eye.  Now  make  the  tube  containing  the  reflector  B 
revolve  within  the  other,  so  that  that  reflector  shall  revolve  round  the  ray  A  B  as  an  axis,  preserving  the  same 
inclination.  Then  will  the  twice  reflected  ray  revolve  with  equal  ai'gular  motion,  and  describe  a  conical 
surface.  But  in  so  doing,  it  will  be  observed  to  vary  in  intensity,  and  at  two  points  of  the  revolution  of  the  tube 
B  will  disappear  altogether.  Now  if  we  attend  to  the  position  of  the  reflectors  at  this  moment,  it  will  be  found 
that  the  planes  of  the  first  and  second  reflexion  make  a  right  angle. 

827.  By  repeating  these  experiments  with  all  sorts  of  reflecting  media,  and  determining  by  exact  measurement  the 
angles  at  which  the  original  ray  must  be   incident    that   polarization  shall  take  place,  and  those  at  which  a 
polarized  ray   ceases   to  be    reflected,  the  following  laws  have  been  ascertained   to  hold   good,  previous  to 
announcing  which  a  definition  will  be  necessary. 

828.  Definition.     The   plane   of  polarization  of  a  polarized  ray  is  the   plane  in  which  it  must  have   undergone 
Plane  of       reflexion,  to  have  acquired  its  character  of  polarization  ;  or  that  plane  passing  through  the  course  of  the  ray 
polarization  perpendicular  to  which  it  cannot  be  reflected  at  the  polarizing  angle  from  a  transparent  medium  ;  or,  again,  that 

plane  in  which,  if  the  axis  of  a  tourmaline  plate  exposed  perpendicularly  to  the  ray  be  situated,  no  portion  of 
the  ray  will  be  transmitted.  Also,  a  polarized  ray  is  said  to  be  polarized  in  its  plane  of  polarization,  as  just 
defined. 

829.  The  plane  of  polarization  of  any  polarized  ray  is  to  be  considered  as  one  of  the  sides  of  the  ray  which  thus, 
Sides  of  a     in  all  its  future  progress,  carries  with  it  certain  relations  to  surrounding  fixed  space,  which   must  be  regarded, 
polarized      while  they  continue  unchanged,  as  inherent  in  the  ray  itself,  and  as  having  no  further  any  relation  to  the  parti- 
cular mode  in  which  they  originated. 

830.  The  laws  of  polarization  by  reflexion  are  these  : 

Laws  of  po-       Law  1.     All  reflecting  surfaces  are  capable  of  polarizing  Might  if  incident  at  proper  angles;    only,  metallic 

larizationby  bodies,  or  bodies  of  very  high  refractive  powers,  appear  to  do  so  but  imperfectly,  the   reflected    ray  not  entirely 

flexion.      disappearing  in  circumstances  when  a  perfectly  polarized  ray  would  be  completely  extinguished.     Of  this  more 

hereafter. 

§31.  Law  2.     Different  media  differ  in  the   angles  of  incidence  at  which  they  polarize  light ;  and  it  is  found,  that 

Law  2.         these  angles  may  always  be  determined  from  the  following  simple  and  elegant  relation,  discovered  by  Dr.  Brewster 
Brewster's    after  a  laborious  examination  of  an  infinite  variety  of  substances, 
law  of  the         ffa  tangmt,  of  the  polarizing  angle  for  any  medium  is  the  index  of  refraction  belonging  to  that  medium. 

Thus,  the  indices  of  refraction  of  water,  crown-glass,  and  diamond,  being  respectively  1.336,  1.535,  and  2.487, 
their  respective  polarizing  angles  will  be  53°  11',  56°  55',  and  68°  6'.  For  diamond,  however,  or  bodies  of  very 
high  refractive  powers,  we  must  understand  by  the  polarizing  angle,  that  angle  of  incidence  at  which  the  reflected 
ray  approximates  most  nearly  to  the  character  of  a  ray  completely  polarized. 

832  I'  f°"ows  from  this  law,  that  one  and  the  same  medium  does  not  polarize  all  the  coloured  rays  at  the  same 

A!]  the  '       angle,  and  that  therefore  the  disappearance  of  the  reflected  pencil  can  never  be  total,  except  where  the  incident 
colours  not   ray  is  homogeneous.     This  will  account  in  some  degree  for  the  want  of  complete  polarization  oi  a  white   ray, 
polarized  at  reflected  at  any  angle  from  highly  refractive  media,  which  are  generally  also  highly  dispersive.     Of  the  reality 
of  the  fact,  it  is  easy  to  satisfy  oneself  by  a  very  simple  experiment,  which  we  have  often  made.     Receive  a  sun- 
beam  on  a  plane  glass,  with   the  back  roughened   and  blackened,  at  an    incidence  (<?)  nearly  equal  to   the 
polarizing   angle  (a,)  and  let  the  reflected   ray  pass  into  a  darkened  room,  and  fall  on   another  similar   glass, 
v.hich  may  be  held  in  the  hand,  so  as  to  reflect  the  ray  in  a  plane  at  right  angles  to  that  of  the  first  reflection,  >md 


L  I  G  II  T.  C07 

Light.      also  at  an  angle  (0')  nearly  equal  to  the  polarizing  angle  (a')  of  the  second  plate.     It  will   be  easy  to  fin:!  a      Part  IV. 
— ^— ^-  position  where  the  reflected  ray  (which  must  be  received  on  a  white  screen)  very  nearly  vanishes  ;  but  no  adjust-  ""•"• "-v-""'' 
ment  of  the  angles  of  incidence  0  and  &  will  produce  a  total  disappearance.     When  the  disappearance  is  most  Pr?ve.<l  by 
nearly  total,  the  reflected  light  is  coloured  of  a  neutral  purple  ;   the  yellow,  or  most  luminous  rays,  being  now  exFer""c"u 
totally  extinguished.     In  this  position,  if  0  remain  constant,  and  &  the  incidence  on  the  second  plate  be  varied 
a  little  on  one  side  or  the  other  of  the  polarizing  angle  «',  the  reflected  ray  assumes  on  the  one  hand  a  pretty 
intense  blue-green,  and  on  the   other  a  ruddy  plum  colour  or  amethyst  red.     The  several  changes  of  tint, 
arising  from  variations  of  incidence  on  both  plates,  were  observed  to  be  as  follows  : 

{0'  <  a' ;         Reflected  ray,  Strong  green. 
Intermediate, — -    White. 
0'  >  a' ; •    Pale  red  or  amethvst. 


0'  <  a' ; Strong  blue  green. 

2d.     0  =.  a;  -J    0'  =  a' ;  •  Neutral  purple. 

; Strong  plum  colour. 

; •  Light  greenish  blue. 

3d.     0  >  a ;  -J    Intermediate,  —  White. 

; Strong  red,  or  plum  colour. 


(    0' 

a ;  •<     In 
(.  0' 


The  rationale  of  these  changes  of  colour  will  be  more  evident  when  we  have  announced  the  following  law, 
which  expresses  one  of  the  most  general  and  distinguishing  characters  of  polarized  light. 

Law  3.     When  a  polarized  ray  (no  matter  how  it  acquired  its  polarization)  is  incident  on  a  reflecting  surface       833. 
of  a  transparent,  or  other  medium  capable  of  completely  polarizing  light,  in  a  plane  perpendicular  to  that  of  the  Law  3. 
ray's  polarization,  and  at  an  angle  of  incidence  equal  to  the  polarizing  angle  of  the  medium,  no  portion  of  the  Npti-refiex- 
ray  will  be  reflected.     If  the  medium  be  of  such  a  nature  as  to  be  capable  only  of  incompletely  polarizing  light,  L^^g-i 
a  portion  will  be  reflected,  but  much  less  intense  than  if  the  incident  ray  were  unpolarized.  \-]a^t  ;n 

It  is  evident  that  this  property  may  be  employed  to  distinguish  polarized  from  common  light,  as  well  as  that  of  certain,  and 
extinction  by  a  plate  of  tourmaline.  It  is,  however,  much  less  convenient  though  better  adapted  for  delicate  wnat  cases. 
inquiries. 

The  polarizing  angle  for  white  light  is,  in  fact,  the  angle  for  the  most  luminous  or  mean  yellow  rays  ;  and       83-i. 
when  the  two  reflexions,  in  planes  at  right  angles  to  each  other,  are  made  at  this  angle,  the  yellow  rays  only  Explanation 
totally  escape  reflexion,  but  a  very  small  portion  both  of  the  red  and  blue  end  of  the  spectrum  are  reflected,  and  of  thet 
form  a  feeble  purple  beam,  such  as  above  described.     The  polarizing  angle  for  red  rays  being  less  than  for  violet,  Jj^"^; '" 
it  is  evident  that  when  either  0  or  0'  is  equal  to  the  polarizing  angle  for  red,  it  will  be  less  than  that  of  yellow,  experiment. 
and  still  less  than  that  of  blue  and  violet  rays ;  thus,  the  red  disappears  most  completely  from  the  reflected  beam 
in  those  cases  when  0  or  ff  are  less  than  a  or  a',  leaving  an  excess  of  the  green  and  blue  rays,  and  vice  versa  in 
the  converse  cases.     Thus,  too,  if  0  be  •<  a,  and  at  the  same  time  O1  <  a,  the  colour  produced  will  be  a  more 
intense  green  than  if  the  incidences  deviated  opposite  ways  from  the  polarizing  angles ;  and  it  is  evident,  that 
a  compensation  may  arise  from  the  effect  of  such  opposite  deviations  giving  an  intermediate  white  ray,  exactly 
as  we  see  to  have  happened. 

Some  very  remarkable  consequences  follow  from  the  law  announced  by  Dr.  Brewster  for  finding  the  polarizing       835. 
angle,  which  may  be  presented  in  the  form  of  distinct  propositions.     Thus, 

Prop.  1.     When  a  ray  is  incident  on  a  transparent  surface,  so  that  the  reflected  portion  shall  be  completely       836. 
polarized,  the  reflected  and  refracted  portions  make  a  right  angle.     For  0  being  the  angle  of  incidence,  we  have  Consequen- 

,,        ••     n  ces  °^  l'lc 

tan  p  =  n  and  />,  being  the  angle  of  refraction,   sin  a  =  ^—  =  ^—  =  cos  0.     Therefore  p  —  90°  -  0,  but  0  1?w,?f  Pol;" 

p.          tan  9  nzaUon. 

being  the  angle  of  incidence  is  also  that  of  reflexion,  and  p  -f-  0  is  therefore  equal  to  the  supplement  of  the 
angle  between  the  reflected  and  refracted  rays,  which  is  therefore  a  right  angle.  Q.  E.  D. 

Prop.  2.     When  a  beam  of  common  light  is  incident  at  the  polarizing  angle  on  a  parallel  plate  of  a  transparent      837. 
medium,  not  only  the  portion  reflected  at  the  first  surface,  but  also  that  reflected  internally  at  the  second,  and  Polarization 
the  compound  reflected  ray,  consisting  of  both  united,  are  polarized.  by  internal 

Since  sin  p  =  cos  0,  and  since  p  is  also  the  angle  of  incidence  on  the  second  surface,  we  shall  have  tan  p  =  re 

cotan  0  = =  —  =  index  of  refraction  oulof  the  medium.     Hence,  a  is  the  angle  of  polarization  for  rays 

tan  0        p. 

'nternally  incident,  and  therefore  that  portion  of  the  beam  which,  having  penetrated  the  first  surface,  falls  on  the 
second,  being  incident  at  its  polarizing  angle,  the  portion  reflected  here  will  also  be  polarized,  and  being  again 
incident  on  the  first  surface,  in  the  plane  of  its  polarization,  that  part  of  it  which  is  transmitted  will  (as  we  shall 
see  hereafter)  suffer  no  change  in  its  plane  of  polarization,  so  that  both  it  and  the  first  reflected  ray  will  come 
off  polarized  in  the  same  plane.  Q.  E.  D. 

Carol.  1.     Hence,  to  obtain  a  stronger  polarized  ray,  we  may  dispense  with  roughening  or  blackening  the       838. 
posterior  surface,  provided  we  are  sure  that  the  surfaces  are  truly  parallel. 

If  a  series  of  parallel  plates  be  laid  one  on  the  other  so  as  to  form  a  pile,  the  portions  reflected  from  the       ,     ^ 
several  surfaces  all  come  off  polarized  in  the  same  plane,  and  by  this  means  a  very  intense  polarized  ray  may  be  ,,^1*;° 
obtained.     It  can  never,  however,  for  a  reason  we  shall  presently  state,  contain   more  than  half  the  incident  an  intenw 
light,  whatever  be  the  number  of  plates  employed.  polarized 

3  u  2  beam. 


508  LIGHT 

Light.          For  a  great  variety  of  optical  experiments,  a  pile  consisting  of  ten  or  a  dozen  panes  of  common  window-glass 

>—""v-»>'  set  in  a  frame,  is  of  great  use  and  very  convenient.     Such  a  pile  laid  down  before  an  open  window  affords  a 

840.      dispersed  beam,  each  ray  of  which  is  polarized  at  the  proper  angle,  and  of  great  intensity  and  very  proper  for 

the  exhibition  of  many  of  the  phenomena  hereafter  to  be  described. 

"•  Prop.  3.     If  a  ray  be  completely  polarized  by  reflexion  at  the  surface  of  one  medium,  and  the  reflected  ray 

completely  transmitted  or  absorbed  at  that  of  a  second,  Required   the   inclination  of  the  two   surfaces  to  each 
other  ? 

Let  a  and  uf  be  the  polarizing  angles  of  the  respective  media ;  then,  since  the  planes  of  reflexion  are  at  right 
angles  to  each  other,  and  a,  a'  are  the  angles  of  incidence,  if  we  call  I  the  inclination  required,  we  shall  have  by 
Art.  104,  cos  I  =  cos  a  .  cos  a'.  Now,  if  p,  p!  be  the  refractive  indices  of  the  media,  we  have  tan  a  =r  p, 
tan  a'  =  u',  and  therefore 

tan  I  =   vy  +  pH  -j_  ^  //? 
S42  Carol.  1.     If  the  media  be  both  alike, 

tan  I  =  p  .  A/2  -j-  /•* ;  or  cos  I  =  ^. 

Thus,  in  the  case  of  crown-glass,  ft  =  1.535  and  I  =  72°  40',  as  in  Art.  825. 

S43.  By  the  help  of  this  law,  connecting  the  angle  of  polarization  with  the  refractive  index,  we  may  easily  deduce 

Method  of  the  one  from  the  other.  This  affords  a  valuable  and  ready  resource  in  cases  to  which  other  methods  can  hardly 
refractive""  ^e  aPP''e^'  *°r  ascertaining  the  refractive  powers  of  media,  which  are  either  opake,  or  in  such  small  or  irregularly 
indices  bv  snaPed  masses,  that  they  cannot  be  used  as  prisms.  For  ascertaining  the  angle  of  polarization,  only  one 
polarization,  polished  surface,  however  small,  is  necessary,  and  we  have  only  to  receive  a  ray  reflected  from  it  on  a  blackened 
glass,  or  other  similar  medium  of  known  refractive  index,  at  the  polarizing  angle,  and  in  a  plane  perpendicular 
to  that  at  which  it  is  reflected  by  the  surface  under  examination.  For  this  purpose  it  is  convenient  to  have  the 
glass  plate  (or,  which  is  better,  a  polished  plate  of  obsidian  or  dark  coloured  quartz)  set  in  a  tube  diagonally, 
so  as  to  reflect  laterally  the  ray  which  traverses  the  axis  of  the  tube.  At  the  other  end,  the  substance  to  be 
examined  must  be  fixed  on  a  revolving  axis  perpendicular  to  the  axis  of  the  tube,  and  having  its  plane  adjusted 
so  as  to  be  parallel  to  the  former,  which  must  then  be  turned  round  till  the  dispersed  light  of  the  clouds, 
reflected  by  it,  is  entirely  extinguished  by  the  obsidian  plate,  and  the  inclination  of  the  reflecting  surface  to  the 
axis  of  the  tube  in  this  situation  may  be  measured  by  a  divided  circle,  connecting  with  the  axis  of  rotation.  By 
this  means  we  may  ascertain  the  polarizing  angles,  and  therefore  the  refractive  indices  of  the  smallest  crystals, 
or  of  polished  stones,  gems,  &c.,  set  in  such  a  manner  as  not  to  admit  of  other  modes  of  examination.  To 
insure  a  fixed  zero  point  on  the  graduated  circle,  the  following  mode  (among  many  others)  may  be  resorted  to. 
Let  a  polished  metallic  reflector  or  small  piece  of  looking-glass  be  permanently  attached  to  the  revolving  axis,  so 
that  its  plane  shall  be  perpendicular  to  the  axis  of  the  tube,  when  the  index  of  the  divided  circle  marks  0°  0'.  This 
adjustment  being  made  once  for  all,  let  the  surface  to  be  examined  be  attached  by  wax  or  otherwise,  not  to  the 
axis  itself,  but  to  a  ring  turning  stiffly  on  it.  Then,  bringing  the  image  of  the  sun,  or  any  very  distant  object, 
sufficiently  bright  or  well  defined,  seen  in  the  reflector,  to  coincide  with  any  other  equally  well  defined,  and 
also  at  a  great  distance,  alter  the  attachment  of  the  substance  by  pressure  on  the  wax,  and  by  turning  round  the 
ring,  till  a  similar  coincidence  is  obtained  when  the  eye  is  transferred  to  it.  Then  we  are  assured  that  the  two 
surfaces  are  parallel,  and  that  therefore  the  reading  off  on  the  circle  measures  the  true  angle  between  the  axis 
of  the  tube  and  the  perpendicular,  or  the  angle  of  reflexion,  or  at  least  differs  from  it  only  by  a  constant 
quantity,  which  may  be  ascertained  at  leisure,  and  applied  as  index  error.  (This  mode  of  bringing  a  movable 
surface  to  a  fixed  position  with  respect  to  the  divisions  of  an  instrument,  is  applicable  to  a  great  variety  of 
cases,  and  is  at  once  convenient  and  delicate.) 

844.  Dr.  Brewster  has  remarked,  that  glass  surfaces  frequently  exhibit  remarkable,  and  apparently  unaccountable, 
Irregular      deviations  from  the  general  law  ;  but  on  minute  examination  he  found  that  this  substance  is  liable  to  a  superficial 

1  tar"'sn'  or  formation  of  infinitely  thin  films  of  a  different  refractive  power  from  the  mass  of  glass  beneath.  As 
tne  P°'ar'zed  ray  never  penetrates  the  surface,  its  angle  of  polarization  is  determined  solely  by  this  film,  which 
is  too  thin  to  admit  of  any  direct  measure  of  its  refractive  index.  When  this  tarnish  has  gone  to  a  great  extent, 
scales  of  glass  detach  themselves,  as  is  seen  in  very  old  windows,  (especially  those  of  stables,)  and  even  in  green 
glass  bottles  which  have  long  lain  in  damp  situations,  and  which  acquire  a  coat  actually  capable  of  being  mistaken 
for  gilding. 

845.  In  metallic  or  adamantine  bodies,  which  polarize  light  but  imperfectly,  that  angle  at  which  the  reflected  beam 
Action  of      approaches  nearer  in  its  character  to  those  described  as  of  polarized  light,  is  to  be  taken  for  the  angle  of  pola- 
rization, and  from  it  the  refractive  power  may  still  be  found.     The  results   deduced  by  this  means  for  metallic 
bodies,  agree  with  those  obtained  from  the  quantity  of  light  reflected,  in  assigning  very  high  refractive  powers  to 
them.     Thus,  for  steel  the  polarizing  angle  is  found  to  be  above  71°,  and  for  mercury  76'£°,  and  their  indices  of 
refraction  are,  therefore,  respectively  2.85  and  4.16.     This  latter  result,  indeed,  differs  greatly  from  that  of  Art. 
594,  but  the  observations  are  so  uncertain,  and  the  angle  of  greatest  polarization  so  indefinitely  marked,  (not 
to  mention  the   errors  to  which  a  determination  of  tjie  reflective  power  itself  is  liable   to,)   that  we   cannot 
expect  coincidence  in  such  determinations.     Perhaps  5.0  may  be  taken  as  a  probable  index. 

846.  The  law  of  polarization  announced  by  Dr.  Brewster  is  general,  and  applies  as  well  to  the  polarization  of  light 
at  the  separating  surfaces  of  two  media  in  contact,  as  at  the  external  or  internal  surface  of  one  and  the  same 
medium.     He  has  attempted  to  deduce  from  it  several  theoretical  conclusions,  as  to  the  extent  and  mode  of 
action  of  the  reflecting  and  refracting  forces,  for  which  we  must  refer  the  reader  to  his  Paper  on  the  subject 
Philosophical  Transactions,  1916 


LIGHT.  509 

If  a  ray  be  reflected  at  an  angle  greater  or  less  than  the  polarizing  angle,  it  is  partially  polarized,  that  is  to     Part  IV. 
say,  when  received  at  the  polarizing  angle  on  another  reflecting  surface,  which  is  made  to  revolve  round  the  <»— ~v^— ' 
reflected  ray  without  altering  its  inclination  to  it,  the  twice  reflected  ray  never  vanishes  entirely,  but  undergoes      847. 
alternations  of  brightness,  and  passes  through  states  of  maxima  and  minima  which  are  more  distinctly  marked  Partial  po!a- 
according  as  the  angle  of  the  first  reflexion  approaches  more  nearly  to  that  of  complete  polarization.     The  same  nzati<1"- 
is  observed  when  a  ray  so  partially  polarized  is  received  on  a  tourmaline  plate,  revolving  (as  above  described) 
in  its  own  plane.     It  never  undergoes  complete  extinction,  but  the  transmitted  portion  passes  through  alternate 
maxima  and  minima  of  intensity,  and  the  approach  to  complete  extinction  is  the  nearer  the  nearer  the  angle  of 
reflexion  has  been  to  the  polarizing  angle.    We  may  conceive  a  partially  polarized  ray  to  consist  of  two  unequally  How 
intense  portions  ;  one  completely  polarized,  the  other  not  at  all.  It  is  evident  that  the  former,  periodically  passing  conceived. 
from  evanescence  to  its  total  brightness,  during  the  rotation  of  the  tourmaline  orreflector,  while  the  latter  remains 
constant  in  all  positions,  will  give  rise  to  the  phenomenon  in  question.     And  all  the  other  characters  of  a  par- 
tially polarized  ray  agreeing  with  this  explanation,  we  may  receive  it  as  a  principle,  that  when  a  surface  does  not 
completely  polarize  a  ray,  its  action  is  such  as  to  leave  a  certain  portion  completely  unchanged,  and  to  impress 
on  the  remaining  portion  the  character  of  complete  polarization.     Thus  we   must  conceive  polarization  as  a 
property  or  character  not  susceptible  of  degree,  not  capable  of  existing  sometimes  in  a  more,  sometimes  in  a  less, 
intense  state.     A  single  elementary  ray  is  either  wholly  polarized  or  not  at  all.     A  beam  composed   of  many 
coincident  rays  may  be  partially  polarized,  inasmuch  as  some  of  its  component  rays  only  may  be  polarized,  and 
the  rest  not  so.     This  distinction  once  understood,  however,  we  shall  continue  to  speak  of  a  ray  as  wholly  or 
partially  polarized,  in  conformity  with  common  language.     We  shall  presently,  however,  obtain  clearer  notions 
on  the  subject  of  unpolarized  light,  and  see  reason  for  discarding  the  term  altogether. 

If  a  ray  be  partially  polarized  by  reflexion,  Dr.  Brewster  has  stated  that  a  second  reflexion  in  the  same  plane      848. 
renders  this  polarization  more  complete,  or  diminishes  the  ratio  of  the  unpolarized  to  the  polarized  light  in  the  Polarization 
reflected  beam  ;   and  that  by  repeating  the  reflexion,  the  ray  may  be  completely  polarized,  although  none  of  the  ''?  several 
angles  of  reflexion  be  the  polarizing  angle.     Thus  he  found,  that  one  reflexion  from  glass  at  56°  45'  of  incidence,  re 
two  at  incidences  of  62°  30'  or  at  50°  20',  three  at  65°  33'  or  at  46°  30',  four  at  67°  33'  or  43°  51',  and  so  on, 
alike  sufficed  to  operate  the  complete  polarization  of  the  ray  finally  reflected,  provided  all  the  reflexions  were 
made  in  one  plane.     At  angles  above  82°,  or  below  18°,  more  than  100  reflexions  were   required  to   produce 
complete  polarization. 

§  IV.     Of  the  Laws  of  Reflexion  of  Polarised  Light. 

When  polarized  light  is  reflected  at  any  surface,  transparent  or  otherwise,  the  direction  of  the  reflected  portion      849. 
is  precisely  the  same  as  in  the  case  of  natural  light,  the  angle  of  reflexion  being  equal  to  that  of  incidence ;  the 
laws  we  are  now  to  consider  are  those  of  the  intensity  of  the  reflected  light,  and  of  the  nature  of  its  polarization 
after  reflexion. 

One  essential  character  of  a  polarized  ray  is,  its  insusceptibility  of  reflexion  in  a  plane  at  right  angles  to  that      850. 
of  its  polarization  when  incident  at  a  particular  angle,  viz.  the  polarizing  angle  of  the  reflecting  surface.     In  Intensity  of 
this  case,  the  intensity  I  of  the  reflected  ray  is  0.     In  all  other  cases  it  has  a  certain  value,  which  we  are  now  to  reflection  of 
inquire.     Let  us  suppose,  then,  to  begin  with  the  simplest  case,  that  the  polarized  ray  fails  on  the  reflecting  ray°i indent 
surface  at  a  constant  angle  of  incidence,  equal  to  its  polarizing  angle,  and  that  the  reflecting  surface  is  turned  at  thepola- 
round  the  incident  ray  as  an  axis,  so  that  the  plane  of  incidence  shall  make  an  angle  (=  «)  of  any  variable  mag-  rizing  angle 
nitude  with  the  plane  of  polarization.     It  is  then  observed,  as  we  have  seen,  that  when  a  —  90°  or  270°,  we  have  'n  any  p'*ne. 
1=0,  and  when  a  =r  0°,  or  180°,  I  is  a  maximum.     Hence,  it  is  clear  that  I  is  a  periodic  function  of  «,  and  the 
simplest  form  which  can  be  assigned   to  it  (since  negative  values  are  inadmissible)  is  I  =  A  .  (cos  a)2.     This 
value,  which  was  adopted  by  Malus  on  no  other  grounds  than  those  we  have  stated,  is  however  found  to  represent 
the  variation  of  intensity  throughout  the  quadrant,  with  as  much  precision  as  the  nature  of  photometrical  experi- 
ments admits,  and  we  must  therefore  receive  it  as  an  empirical  law  at  present,  for  which  any  good  theory  of 
polarization  ought  to  be  capable  of  assigning  a  reason  a  priori. 

A  remarkable  consequence  follows  from  this  law.     It  is  that,  so  far  as  the  intensity  of  the  reflected   ray  i*       851. 
concerned,  an  ordinary  or  unpolarized   ray  may  be  regarded  as  composed   of  two  polarized  rays,   of  equal  ^^P0'*" 
intensity,  having  their  planes  of  polarization   at  right  angles  to  each  other.     For  such  a  compound  ray  being  equivalent 
incident  on  a  reflecting  surface,  as  above  supposed,  if  a  be  the  inclination  of  the  plane  of  polarization   of  one  to  two  pola- 
portion  to  that  of  incidence,  90  —  a  will  be  that  of  the  other,  and,  therefore,  since  rized  ones. 

A  .  (cos  a)'  +  A  .  (cos  .  90  -  «)8  =  A,  («) 

the  reflected  ray  will  be  independent  of  a,  and  therefore  no  variation  of  intensity  will  be  perceived  on  turning 
the  reflecting  surface  round  the  incident  ray  as  an  axis,  which  is  the  distinguishing  character  of  unpolarized  light. 
Any  such  pair  of  rays  as  here  described  are  said  to  be  oppositely  polarized. 

When  the  polarized  ray  is  not  incident  at  the  polarizing  angle,  but  at  any  angle  of  incidence,  the  law   of      85?. 
intensity  of  the  reflected  ray  is  more  complicated.     M.  Fresnel  has  stated  the  following  as  the  general  expression  Fresnel's 
for  it.    Let  the  intensity  of  the  incident  ray  be  represented  by  unity,  and  calling,  as  before,  a  the  inclination  of  the  S1™™1  la* 
plane  of  incidence  to  that  of  primitive  polarization,  and  i  the  angle  of  incidence,  i'  the  corresponding  angle  of  jn[enjtv  nf 
refraction.     Then  will  the  intensity  of  the  reflected  ray  be  represented  by  a  reflected 

ray 


510  LIGHT. 

Li§ht-  sin'  (j  -  z')  ,    tan!  (i  -  i')  IVt  IV. 

s— v— -'  I  =     .  .,.   ,    .,.  •  cos8  a  -f  — —  .  sin'  a.  (6)  v^^^, 

sin'  (z  -f-  z')  tan8  (z  -f- 1  ) 

This  formula  is  in  some  degree  empirical,  resulting  partly  from  theoretical  views,  of  which  more  hereafter,  and 
being  not  yet  verified,  or  indeed  compared  with  experiment,  except  in  particular  cases,  by  M.  Arago,  whose 
results,  so  far  as  they  go,  are  consonant  with  it. 

It  will  be  well  to  examine  some  of  these.     And  first,  then,  when  a.  =  90°,  and  i  =  the  polarizing  angle  of  the 
^ular     reflecting  surface,  we  have  by  (835  and  836)  z  +  i'  =  90°,  and  therefore  tail  (i  +  i')  =  oo,  so  that  1  =  0.     In 
examined,    these  circumstances,  then,  the  reflected  ray  is  completely  extinguished,  which  agrees  with  fact. 

854.          2dly.     When  the  incidence  is  perpendicular,  we  have,  in  this  case,  both  i  and  i1  vanishing,  and  each  term  of  I 

larhici-'      takes  the  form  — -.     Now  at  the  limit  we  have  (/*  being  the  refractive  index)  i  =  p  .  i',  and  very  small  arcs  being 

equal    to  their  sines   or  t; 
tangents.     Consequently, 


equal   to  their  sines  or  tangents,  we  have  sin  (z  —  i')  =  i'  (/*  —  1) ;  sin  (i  +  t7)  =  i'  (u  +  1),  and  so  for  the 


which  agrees  with  the  expression  deduced  by  Dr.  Young  and  M.  Poisson,  (Art.  592,)  for  the  intensity  of  the 
reflected  ray  in  the  case  of  unpolarized  light.  And  if  we  regard  the  unpolarized  ray  as  composed  of  two  rays, 
each  of  the  same  intensity,  (=  ^)  polarized  in  opposite  planes,  the  reason  of  the  coincidence  will  be  evident. 

855.  3d.     When  a  =r  0,  or  the  plane  of  polarization  coincides  with  the  plane  of  incidence,  we  have,  in  general, 

_  sin8  (i  —  i') 
~  sin'  (i  +  i')' 

8 56.  4th.     When  a  =  90°,  or  when  the  plane  of  polarization  is  at  right  angles  to  the  plane  of  incidence, 


~  tan'  (i  +  f) 

857.  5th.     When  a  =  45, 

Intensity  of  f  sin2  (j  —  i')          tan"  (i  -  z')~» 

reflexion  of  I  =  4    1     .    „    .. — ; — nr   T Trr: — ; — ^  (~ •  (e) 

natural  light  '    ^SIn*  (*  +  O  tan*  (»  +  O  J 

This  last  is  the  same  result  with  that  which  would  result  from  the  supposition  of  two  equal  ravs  polarized, 
the  one  in,  the  other  perpendicularly  to,  the  plane  of  incidence,  and  each  of  half  the  intensity  with  the  incident 
beam.  It  is  therefore  the  general  expression  for  the  intensity  of  a  ray  of  natural  or  unpolarized  light  reflected 
at  an  incidence  =  i  from  the  surface.  The  expressions  in  Art.  592  apply  only  to  perpendicular  incidences.  We 
are  thus  furnished  very  unexpectedly  with  a  solution  of  one  of  the  most  difficult  and  delicate  problems  of  experi- 
mental Optics.  Bouguer  is  the  only  one  who  has  made  any  extensive  series  of  photometrical  experiments 
on  the  intensity  of  light  reflected  from  polished  surfaces  at  various  angles,  but  his  results  are  declared  by 
M.  Arago  to  be  very  erroneous,  which  is  not  surprising,  as  the  polarization  of  light  was  unknown  to  him,  and  its 
lajvs  might  affect  the  circumstances  of  his  experiments  in  a  variety  of  ways. 

858.  One  only  need  be  mentioned,  as  every  optical  experimentalist  ought  to  be  aware  of,  and  on  his  guard  against 
Polarization  it,  it  is  that  the  light  of  clear,  blue  sky,  is  always  partially  polarized  in  a  plane  passing  through  the  sun,  and  the 
of  the  light   part  from  which   the  light  is  received.     The  polarization  is  most  complete  in  a  small  circle,  having  the  sun  for 
of  the  sky.    jts  pOje>  an(j  jts  ra(j;ug  about  78°,   (according  to   an  experiment  not  very  carefully  made.)     Now  the  semi- 
supplement  of  this  (which  is  the  polarizing   angle)  is  51°,  which  coincides  nearly  with  the  polarizing  angle  of 
water,  (52°  45'.)     Thus  strongly  corroborating  Newton's  theory  of  the  blue  colour  of  the  sky,  which  he  conceives 
to  be  the  blue  of  the  first  order,  reflected  from  particles  of  water  suspended  in  the  air.     Dr.  Brewster  is  the  first, 
we  believe,  who  noticed  this  curious  fact.     But  to  return  to  our  subject. 

859.  When  the  incident  ray  is  only  partially   polarized,  we  may  regard  it  as  consisting  of  two  portions  :  the  one, 
Case  of  a     which  we  shall  represent  by  a,  completely  polarized  in  a  plane,  making  the  angle  a  with  that  of  incidence  ;  the 
ray  partially  /I    —   a\ 

polarized,     other  =  1  —  a  in  its  natural  state,  or,  if  we  please,  composed  of  two  portions!  — - —  I,  one  polarized  in  the 

plane  of  incidence,  and  one  at  right  angles  to  it.     The  intensity  of  the  reflected  portion  of  the  former  is  equal  to 

sin'(z  -  i')  tan2(z  -  z') 

cos'  a  +  a  .  — -^  .  sin*  a, 


•    •  if    ,     •-•>  •  .     ,..    ,     .,.  • 

sin2  (i  +  i)  tan"  (z  -f  z') 

and  that  of  the  latter  will  be  represented  by 

1  -  a   f  sin*  (z  -  z')         tan8  (z  -  i') 


} 


2        (.  sin8  (i  +  i')         tan8  (i  +  z') 
therefore,  their  sum,  or  the  total  reflected  light,  will  be 

sin'  (z  —  z')       1  +  a  .  cos  2  a        tan8  (i  —  ir)      1  -  a  .  cos  2 


sin'  (z  -f  i')    '  2  ~  tan8  (i  -f-  z')   '  "       ~2~ 

The  above  formula,  it  must  be  observed,  apply  only  to  the  case  of  reflexion  from  the  surfaces  of  uncrystallized 
media.  The  consideration  of  those  where  crystallized  surfaces  are  concerned,  cannot  be  introduced  in  this  part 
of  the  subject. 


LIGHT.  511 

When  the  plane  of  reflexion  coincides  with  that  of  the  primitive  polarization  of  the  ray,  the  polarization  is  not    part  IV. 
changed  by  reflexion.     Hence,  at  a  perpendicular  incidence  it  is  unchanged.     But  in  other  relative  situations  \^-^-^j 
of  life  two   planes  above-mentioned,  the  case  is  different,  and  it  becomes  necessary  to  inquire  what  change        860. 
reflexion  produces  in  the  state  and  plane  of  polarization  of  the  ray.     Now  it  is  found,  as  we  have  already  seen,  Position  of 
that  when  the  reflection  takes  place  in  the  plane  of  primitive  polarization,  if  the  incident  ray  be  only  partially  "jj  P?*"*  °J 
polarized,  the  reflected  one  will  be  more  so,  in  that  plane.     But  if  the  incident  ray  be  completely  polarized,  it  JfjS,,  „.*" 
retains  this  character  after  reflexion,  (except  in   one  remarkable  case,)  and  only  the  plane  of  polarization  is  flecled  rav> 
changed.     Now,  according  to  M.  Fresnel,  the  new  plane  of  polarization  will  make  an  angle  with  the  plane  of 
reflexion,  represented  by  ft,  such  that 

cos  (i  +  V) 

tan  ft  = ,.       .,.  •  tan  a. 

cos  (t   -  r) 

According  to  this  formula,  the  plane  of  polarization  coincides  with  the  plane  of  incidence  when  » -f  V  ==  90°.  Now 
this  is  precisely  the  case  when  the  ray  falls  at  the  polarizing  angle  on  the  reflecting  surface.  If  a  —  90°,  or  the  ray 
before  incidence  be  polarized  in  a  plane  perpendicular  to  the  plane  of  incidence,  it  will  continue  to  be  so  after 
reflexion,  since  in  that  case  we  have  tan  ft  =  oo,  or  f)  =  90°. 

The  formula  has  been  compared  by  M.  Arago  with  experiment  only  in  one  intermediate  case,  viz.  when  861. 
a  =  45°,  and  the  coincidence  of  the  results  with  experiment  at  a  great  variety  of  incidences,  and  over  a  range  of 
values  of  ft  from  -f-  38°  to  -  44°,  both  in  the  case  of  glass  and  water,  is  as  satisfactory  as  can  be  desired.  The 
particulars  of  this  interesting  comparison  will  be  found  in  Annales  de  Chimie,  xvii.  p.  314.  It  may  be 
observed  also,  that  these  results  of  M.  Fresnel  support  one  another,  the  latter  being  concluded  from  the  former 
by  considerations  purely  theoretical,  so  that  every  verification  of  the  one  is  also  a  verification  of  the  other. 

When  the  polarized  ray  is  reflected  from  a  crystallized   surface,  the  intensity  of  the  reflected  portion  is  no       862. 
longer  the  same,  but  depends  on  the  laws  of  double  refraction,  in  a  manner  of  which  more  hereafter.     Whether,  Reflexion 
or  how  far,  the  laws  above  stated  hold  good  for  metallic  surfaces,  remains  open  to  inquiry.  tauTzeTsiu- 

faces. 

§  V.    Of  the  Polarization  of  Light  by  ordinary  Refraction,  and  of  the  Laws  of  the  Refraction  of  Polarized  Light. 

When  a  ray  of  natural  or  unpolarized  light  is  transmitted  through  a  plate  of  glass  at  a  perpendicular  incidence,        863. 
it  exhibits  at  its  emergence  no  signs  of  polarization  ;    but  if  the  plate  be  inclined  to  the  incident  ray,  the  trans-  Polarization 
milled  ray  is  found  to  be  partially  polarized  in  a  plane  at  right  angles  to  the  plane  of  refraction,  and  therefore  v  refrac- 
at  right  angles  to  the  plane  of  polarization  of  the  portion  of  the  reflected  ray  which  has  undergone  that  modifi- 
cation.    The  connection  belween  Ihe  polarized  portions  of  the  reflected  and  refracted  pencils  is,  nowever,  still 
more  intimate,  since  M.  Arago  has  shown  by  a  very  elegant  and  ingenious  experiment  that  these  portions  are  Arago 'slaw, 
always  of  equal  intensity.     This  law  may  be  stated  thus  :  Wften  an  unpolarized  ray  is  partly  reflected  at,  and 
partly  transmitted  through,  a  transparent  surface,  the  reflected  and  transmitted  pencils  contain  equal  quantities  of 
polarized  light,  and  their  planes  of  polarization  are  at  right  angles  to  each  other. 

Hence  it  appears,  that  the  transmitted  ray  contains  a  maximum  of  polarized  light,  when  the  light  is  incident       864. 
al  the  polarizing  angle  of  the  medium,  and  this  maximum  is  equal  to  the  quantity  of  light  the  surface  is  capable 
of  completely  polarizing  by  reflexion.     Now  in  all  media  known,  this  is  much  less  than  half  the  incident  light, 
consequently  the  transmitted  portion  can  never  be  wholly  polarized  by  a  single  transmission. 

When  a  ray  is  totally  reflected  at  the  inner  surface  of  a  medium,  there  is  no  transmitted  portion,  an;I  it  is  a       665 
remarkable  coincidence  with  Ihe  above  law,  lhat  in  this  case  the  reflected  beam  contains  no  polarized  porlion 
whatever. 

With  regard  to  Ihe  portion  of  Hghl  which  has  passed  through  the  surface,  and  has  not  acquired  polarization,       866. 
M.  Arago    maintains  that  it  remains  in  the  slate  of  natural  or  totally  unpolarized  light.     Dr.  Brewster,  on  the  Polarization 
other  hand,  concludes  from  his  experiments,  that,  although  not  polarized,  it  has  undergone  a  physical  change,  by  *«eral 
rendering  it  more  largely  susceptible  of  polarization  by  subsequent  transmission  at  the  same  angle.     The  ques-  "blK|ue 
tion,  in  a  theoretical  point  of  view,  is  a  material  one,  and  apparently  very  easily  decided.     The  facility,  however,  ,jon 
is  only  apparent,  and  as  we  have  no  title  to  decide  it  on  the  grounds  of  our  own  experience,  we  shall  content 
ourselves  with  reasoning  on  the  conclusions  to  which  the  two  doctrines  lead.     Let  1  be  the  light  incident  on  the 
first  surface  of  a  glass  plate  at  the  polarizing  angle,  and,  after  transmission  through  both  surfaces,  lei  a  4-  b  be  the 
intensity  of  the  transmitted  beam,  (and  of  course  1  —  a  —  b  that  of  the  reflected,)  and  let  a  be  the  polarized 
portion  and  6  the  unpolarized.     When  a  -)-  b  falls  on  another  plate  at  the  same  angle,  the  portion  a  being  pola- 
rized in  a  plane  perpendicular  to  that  of  incidence,  and  incident  at  the  polarizing  angle,  will  be  totally  trans- 
mitted, and  ita plane  nf  polarization  (as  may  be  proved  by  direcl  experimenl)  in  this  case  undergoes  no  change. 
Hence  the  portion  a  will  be  transmitted  (supposing  no  absorption)  undiminished  through  any  number  of  sub- 
sequent plates.     With  regard  to  the  portion   6,  if  this  be  to  all  intents  and  purposes   similar  to  natural  light,  it 
will  be  divided  by  reflexion  at  the  second  plate  into  two  portions,  the  first  of  which  =  b  .  (1  —  a  —  b)  being 
reflected  wholly  polarized,  and  the  other  =  b  (a  -f-  b)  will  be  transmilted.     Of  ihis,  Ihe  portion  b  a  will  be  pola- 
rized in  a  plane  at  right  angles  to  that  of  refraction,  and  will  therefore  be  afterwards  transmitted  undiminished 
through  all  the  subsequenl  plates.     But  the  portion  b-  will  be  unpolarized  light,  and  will  be  again  divided  by 
Ihe  Ihird  plate,  and  so  on.     Thus,  there  will  be  ultimately  transmitted  a  pencil,  consisting  of  a  polarized  portion 


512  LIGHT. 

I>ht                                                                       1—6* 
v_  °-»_-  =a+6fl-(-6sa-(-  ....  6""'  a—a. -,  and  an  unpolarized  portion  =  6",  so  that  no  finite  number  of  v 

plates  could  ever  compMely  polarize  the  whole  transmitted  beam. 

867.  On  the  other  hand,  if  the  unpolarized  portion  6  of  the  transmitted  beam  a  +  6  be  more  disposed  than  before, 
Dr.^Brew-    as  Dr.  Brevvster  conceives,  to  subsequent  polarization,  the  progression  above  stated,   instead  of  converging 

'  theory  according  to  the  law  of  a  geometric  progression,  will   converge  more  rapidly,  or  may  even  suddenly  terminate 
polaHzatinn  unc'er  certain  physical  conditions.     Now,  Dr.  Brewster  states  it  as  a  general   law,  deduced  from  his  own  experi- 
Hrewster's    rnents,  that  If  a  pencil  of  light  be  incident  on  a  number  of  uncrystallized  plates,  inclined  at  the  same  or  different 
eeneral  law.  angles,  but  all  their  surfaces  being  perpendicular  to  the  plane  of  the  first  incidence,  the  total  polarization  of  the 
transmitted  pencil  will  commence  when  the  sum  of  the  tangents  of  the  angles  of  incidence  on  each  plate  is  equal 
to  a  certain  "  constant  quantity  due  to  the  refractive  power  of  the  plates,  and  the  intensity  of  the  incident  pencil!1 
This  last  phrase,  which  makes  the  number  and  position  of  the  plates  necessary  to  operate   total  polarization, 
depend  on  the  intensity  of  the  incident  light,  shows  evidently  that  the  total  polarization  here  understood,  is  not 
mathematically,  but  only  approximative!)-  total.     In  fact,  he  states,  this  constant  quantity  for  crown  glass  plates, 
and  for  tin-  flame  of  a  wax  candle  at  \0feet  distance,  to  be  equal  to  the  number  41.84.     In  other  words,  the 
remainder  of  unpolarized  light  for  this  intensity  of  illumination,  becomes  insensible.     Considered  in  this  light, 
we  regard  Dr.  Hrewster's  experiments  as  by  no  means  incompatible  with  the  law  of  decrease  indicated  by  the 
geometric  progression   above-mentioned   and  the  contrary  sense  which  has  been  put  upon  this  expression  by 
M.  Arago,  or  his  commentator,  (Encyciop.  Brit.  Supp.,  vol.  vi.  part  2,  Polarization  of  Light,)  appears  to   us 
strained  beyond  what  strict  criticism  authorizes. 

Conceiving,  then,  as  we  do,  that  no  decided  incompatibility  in  matter  of  fact  exists  between  the  statements  of 
these  distinguished  philosophers,  we  cannot  but  regard  as  most  simple,  that  doctrine  which  recognises  no  change 
of  physical  character  in  the  unpolarized  portion  of  either  the  transmitted  or  reflected  beam.  (See  Art.  848.) 

868.  In  what  has  been  above  said  of  the  polarization  of  the  transmitted  ray,  we  have  not  taken  into  consideration 
Internal        that  part  of  the  light  reflected  at  each  surface  which  is  reflected  back  again,  and  traversing  (partially  at  least)  all 
reflexions     the  plates,  mixes  with  the  transmitted  beam,  and,  being  in  an  opposite  plane,  destroys  a  part  of  its  polarization. 

If  a  pile  of  parallel  glass  plates  be  exposed  to  a  polarized  ray,  so  that  the  angle  of  incidence  be  equal  to  the 
869        polarizing  angle,  and  then  turned  round  the  ray  as  an  axis  preserving  the  same  inclination,  the  following  pheno- 

I'henom'ena  mena  take  Place  : 

of  piles  of  1.  When  the  plane  of  incidence  is  at  right  angles  to  that  of  the  raj's  polarization,  the  whole  of  the  incident 
plates  ex-  light  is  transmitted,  (except  what  is  destroyed  by  absorption  within  the  substance  of  the  glass,  or  lost  by  irregular 
posed  to  reflexion  from  the  inequalities  in  the  surface  arising  from  defective  polish,)  and  this  holds  good  whatever  be  the 
lij-ht""  number  of  the  plates.  The  polarization  of  the  transmitted  ray  is  unaltered. 

2.  As  the  pile  revolves  round  the  incident  ray  as  an  axis,  a  portion  of  the  light  is  reflected,  and  this  increases 
till  the  plane  of  incidence  is  coincident  with  the  plane  of  primitive  polarization,  when  the  reflected  light  is  a 
maximum.  Now,  M.  Arago  assures  us,  that  the  quantity  of  polarized  light  reflected  from  each  plate  is  greater  in 
proportion  to  the  intensity  of  the  incident  beam  than  if  natural  light  had  been  employed ;  and  the  same  pro- 
portion holding  good  at  each  plate,  the  transmitted  ray,  however  intense  it  may  have  been  at  first,  will  be 
weakened  in  geometrical  progression  with  the  number  of  plates,  and  at  length  will  become  insensible  ;  so  that 
in  this  situation  the  pile  will  present  the  phenomenon  of  an  opaque  body.  In  this  reasoning,  the  light  reflected 
backwards  and  forwards  between  the  plates  is  neglected  ;  but  as  it  is  all  polarized  in  the  same  plane,  and  as  in 
this  situation  the  reflexions,  however  frequent,  produce  no  change  in  its  plane  of  polarization,  all  the  reflected 
rays  are  in  the  same  predicament ;  and,  supposing  the  number  of  plates  very  great,  the  total  extinction  of  the 
transmitted  light  will  ultimately  (though  less  rapidly)  take  place. 

870.  Hence,  a  pile  of  a  great  number  of  glass  plates  inclined  at  an  angle  equal  to  the  complement  of  the  polarizing 
Phenomena  angle  (35°  i)  to  a  polarized  ray  ought  to  present  the  same  phenomenon  with  a  plate  of  tourmaline  cut  parallel 
of  piles  of   to  the  axis  of  its  primitive  rhomboid,  alternately  transmitting  and  extinguishing  the  whole  of  the  light  in  the 
oHournra1   success've  quadrants  of  its  rotation,  and  being  thus  either  opaque  or  transparent,  according  to  its  position.     The 
line  plates    analogy,  however,  cannot  fairly  be  pushed  farther,  so  as  to  deduce  from  this  principle  an  explanation  of  the  phe- 
compured.    nomena  of  the  tourmaline  ;  for,  although  it  be  true  that  a  plate  of  tourmaline  so  cut,  is  composed  of  lamina? 

inclined  to  its  surface,  these  laminae  are  in  optical  contact ;  and,  moreover,  their  position  with  respect  to  .the 
surface  is  not  the  same  in  plates  cut  in  all  directions  around  the  axis,  because  although  an  infinite  number  of 
plates  may  be  cut  containing  the  axis  of  a  rhomboid  in  their  planes,  only  three  can  have  the  same  relation  to  its 
several  faces,  parallel  to  which  the  component  laminas  must  be  supposed  to  lie.  Moreover,  the  phenomena  are 
not  produced,  unless  the  tourmaline  be  coloured.  The  analogy  between  piles  of  glass  plates  and  laminae  of  agate 
(of  which  more  presently)  is  also,  we  are  inclined  to  think,  more  apparent  thun  real. 

871.  A  pile  of  plates  such  as  described  above  presents,  moreover,  the  same  difference  of  phenomena  when  exposed 
Furtlier        to  polarized  and  unpolarized  light,  that  a  plate  of  tourmaline  does  ;   since  in  the  latter  case,  supposing  the  pile 
analogy,       sufficiently  numerous,  one  half  the  incident  light  is  transmitted,  completely  polarized  in  a  plane  perpendicular  to 

that  of  incidence. 

872.  The  laws  which  regulate  the  polarization  of  a  pencil  transmitted  by  a   transparent   surface,  inclined  at  any 
proposed  angle  to  the  incident  ray,  and  in  .any  plane  to  that  of  its  primitive  polarization  (supposing  it  polarized) 
remain  open  to  experimental  investigation. 


L  I  G  H  T.  513 

Light.  Part  IV. 

§  VI.  Of  the  Polarization  of  Light  by  Double  Refraction. 


When  a  ray  of  natural  light  is  divided  into  two  by  double  refraction,  in  such  a  manner  that  the  two  pencils  at      873. 
eir  final  emergence  remain  distinct  and  susceptible  of  separate  examination,  they  are  both  found  completely 
polarized,  in  different  planes,  exactly,  or  nearly,  at  right  angles   to  each  other.     To  show   this,  take   a  pretty 


their  final  emergence  remain  distinct  and  susceptible  of  separate  examination,  they  are  both  found  completely  Light  pola- 
polarized,  in  different  planes,  exactly,  or  nearly,  at  right  angles   to  each  other.     To  show   this,  take   a  pretty  "zed  ^ 
thick  rhomboid   of  Iceland    spar,   and,    covering   one  side   of    it  with    a  blackened    card,   or   other   opaque  r(!fja(.[;orl 


(a(.o 

thin  substance,  having  a  small  pinhole  through  it,  hold  it  against  the  direct  light  of  a  window  or  a  candle,  with  oppositely 
the  covered  surface  from  the  eye.     Two  images  of  the  pinhole  will  then  be  seen  :  one,  undeviated  from  the  line  in  the  two 
joining  the  eye  and  the  real  hole,  by  the  ordinarily  refracted  rays  ;   and  the  other,  deviating  from  that  line,  in  a  Pf"0'1* 
plane  parallel  to  the  principal  section  of  the  surface  of  incidence,  by  the  extraordinary.     These  images  will  K! 
appear,  to  the  naked  eye,  of  equal  brightness;  but,  if  we  interpose  a  plate  of  tourmaline,  (as  already  described,)  proof 
and  turn  the  latter  about  in  its  own  plane,  they  will  be  rendered  unequal,  and  will  appe.ir  and  vanish  alternately  thereof. 
at  every  quarter  revolution  of  the  tourmaline  ;  the  ordinary  image  being  always  at  its  maximum  of  brightness, 
and  the  extraordinary  one  extinct,  when  the  axis  of  the  tourmaline  plate  is  perpendicular  to  the  principal  section 
of  the  surface  of  incidence,  and  vice  versd  when  parallel  to  it. 

The  same  thing  happens,  when,  instead  of  examining  the  two  images  through  a  tourmaline  plate,  we  receive       874. 
their  light  on  a  glass  plate  inclined  at  the  polarizing  angle  to  it,  and  turn  this  plate  round  the  ordinary  ray  Expeiimcnt 
as  an   axis.     The  images  will  appear  and  disappear  alternately,  as  the  reflector  performs  successive  quadrants  varitd. 
of  its  revolution. 

Hence,  we  see  that  the  two  pencils  are  completely  and  oppositely  polarized  ;  the  ordinary  pencil  in  a  plane       875. 
passing  through  the  axis  of  the  rhomboid  ;  the  extraordinary  one  in  a  plane  at  right  angles  to  it. 

The  same  phenomenon  is  much  better  seen  by  using  a.  prism  of  any  double  refracting  crystal,  having  such  a      876. 
refracting  angle  as  to  give  two  distinctly  separated  images  of  a  distant  object,  (as  a  candle.)     These  appear  and  ^notller 
disappear  alternately  at  quarter  revolutions  of  a  tourmaline  plate  or  glass  reflector,  and  are  of  equal  brightness  experiment 
at  the  intermediate  half-quarters. 

Double  refraction,  then,  polarizes  the  two  refracted  pencils  oppositely,  into  which  an  unpolarized  incident  ray      877. 
is  separated.     Let  us  now  see  what  happens  to  a  polarized  ray.     For  this  purpose  let  a  plate  of  glass  be  laid  Transmis- 
down  before  an  open  window,  so  as  to  polarize  the  reflected  light,  and  hold  the  rhomboid  of  Iceland  spar  S1°"  °f 
(covered  us  before)  witli  the  covered  side  from  the  eye,  not  (as  in  the  former  experiment)  against  the  direct  light,  1°^"" 
but  inclined  downwards,  against  the  reflected  light  from  the  glass.     Then,  generally  speaking,  two  images  of  through 
the  pinhole  will  be  seen,  but  of  unequal  intensities  ;    and,  if  we  turn  round  the  rhomboid,  in  the  plane  of  the  doubly 
covered  side,  these  images  will  be.  seen  to  vary  perpetually  in  their  relative  brightness,  the  one  increasing  to  a  max-  refracting 
imum,  while  the  other  vanishes  entirely,  and  so  on  reciprocally.     When  the  principal  section  of  the  rhomboid  is  in  media- 
the  plane  of  reflexion  (i.  e.  of  polarization)  of  the  incident  ray,  the  ordinary  image  is  a  maximum  ;  the  extra- 
ordinary is  extinct,  and  vice  versd  when  these  two  planes  make  a  right  angle.     The  experiment  may  be  advan- 
tageously varied  by  using  a  doubly  refracting  prism  ;  and,  while  looking  through  it  at  the  polarized  image  of  a 
candle,  turning  it  round  slowly  in  the  plane  bisecting  its  refracting  angle. 

This  experiment  leads  us  to  the  following  remarkable  law,  vis.  that  if  a  ray,   at  its   incidence   on   a  doubly      878. 
refracting  surface,  be  polarized  in  the  plane  parallel  to  the  principal  section,  it  will  not  suffer  bifurcation,  but  Unequal 
will  pass  wholly  into  the  ordinary  image  ;  if,  on  the  other  hand,  its  plane  of  primitive  polarization  be  perpen-  di!VIp0v  of 
dicular  to  the  principal  section,  it  will  pass  entirely  into  the  extraordinary  image.     In  intermediate  positions   of  be^v'ifen 
the  plane  of  primitive  polarization,  bifurcation  takes  place,  and  the  ray  is  unequally  divided  between  the   two  the  two 
refracted  pencils,  in  every  case  except  when  the  plane  of  primitive  polarization  makes  an  angie  of  45°  with  the  refracted 
principal  section.     In  general,  if  a  be  the  angle  last  mentioned,  and  A  the  incident  light,  (supposing  none  lost  Pencils. 
by  reflexion,)  A  .  cos2  a  will  be  the  intensity  of  the  ordinary,  and  A  .  sin2  a  of  the  extraordinary  pencil,  their 
sum  being  A. 

All  these  changes  and  combinations  are  exhibited  in  the  following  remarkable  experiment  of  Huygens,  which,      879. 
reasoned  on  by  himself  and  Newton,  first  gave  rise  to  the  conception  of  a  polarity,  or  distinction  of  sides,  in  the  Huygens's 
rays  of  light  when  modified   by  certain  processes.     Take  two  pretty  thick  rhomboids  of  Iceland  spar,  (which  exPcrl 
should  be  very  transparent,  as  they  are  easily  procured,)  and  lay  them  down  one  upon  the  other,  so  as  to  have 
their  homologous  sides  parallel,  or  so  that  the  molecules  of  each  shall  have  the  same  relations  of  situation  as  if 
the  two  rhomboids  were  contiguous  parts  of  one  larger  crystal.     They  should  be  laid  on  a  sheet  of  white  paper 
having  a  small,  very  distinct,  and  well-defined  black  spot  on  it.     This  spot  then  will  be  seen  double  through  the 
combined  crystals,  as  if  they  were  one,  (a,  fig.  173,)  and  the  line  joining  the  images  will  be  parallel   to  the  Fig.  137. 
principal  section  of  either.     Now,  let  the  upper  crystal  be  turned  slowly  round  in  a  horizontal   plane  on  the 
lower,  and  two  new  images  will  make  their  appearance  between  the  two  first  seen,  which,  at  first,  are  very  faint, 
as  at  b,  fig.  173,  and  form  a  very  elongated  rhombus  with  the  two  former.     They  increase,  however,  in  intensity, 
while  the  other  pair  diminishes,  till  the  angle  of  rotation  of  the  upper  crystal  is  45°,  where  the  appearance  of  the 
images  is  as  at  c.     Continuing  the  rotation,  the  rhomb  approaches  to  a  square,  as  at  d,  and  the  two  original  images 
have  become  extremely  faint  ;  and  when  the  rotation  is  just  90°,  they  will  have  disappeared  altogether,  leaving 
the  others  diagonally  placed,  as  at  e.     As  the  rotation  still  proceeds,  they  reappear  and  increase  in  brightness,  till 
the  angle  of  revolution  =  90°  -j-  45°  ==  135°,  when  the  images  are  all  equal,  as  at  f;  after  which  the  original 
images  still  increasing,  and  the  others  diminishing,  the  appearance  g  is  produced,  which,  on  the  completion  of 
a  precise  half  revolution,  passes  into  h  by  the  union  of  both  the  original  images  into  one,  and  the  total  evanes- 
VOL.  IT.  3  x 


514 


LIGHT. 


880. 


881. 
Use  of  an 

achromatic 

refracting 
prism 


Fig.  174. 
First  achro 
matized  by 


882. 

Dr.  Wollas- 
ton's  mode 


•-ion  of 
images. 
Fig.  175. 


883. 
Action  of 

s. 

nTefouble 
refraction, 


cence  of  the  other  pair.  In  this  case,  only  single  refraction  (apparently)  happens;  or,  rather,  the  double  refrac- 
"  tions  of  the  two  rhomboids  taking-  place  in  opposite  directions,  and  being  equal  in  amount,  compensate  each 
other.  Unless,  however,  the  rhomboids  be  of  exactly  equal  thickness,  this  precise  compensation  will  not  take 
place,  and  the  images  will  remain  distinct,  though  at  a  minimum  of  distance.  We  may  express  the  four  images 
thus: 

O  o,  the  image  ordinarily  refracted  by  both  rhomboids. 

O  e,  the  image  refracted  ordinarily  by  the  first,  and  extraordinarily  by  the  second. 

E  o,  the  image  refracted  extraordinarily  by  the  first,  and  ordinarily  by  the  second. 

E  e,  the  image  refracted  extraordinarily  by  both. 

Then,  if  A  be  the  intensity  of  the  incident  light,  supposing  none  lost  by  reflexion  or  absorption, 

O  o  =  \  A  .  cos2  a  =  E  e;     O  e  =  £  A  .  sin2  a  =  E  o, 
and  the  sum  of  all  the  four  images  =  A. 

The  same  phenomena  (with  some  unimportant  variations)  take  place  when  we  apply  two  doubly  refracting 
prisms  one  behind  the  other  close  to  the  eye,  and  view  a  distant  object  through  them,  turning  one  round  on 
the  other.  The  rationale  of  these  phenomena  follows  so  evidently  from  the  laws  stated  in  Art.  875  and  878, 
that  it  will  not  be  necessary  to  enlarge  on  it. 

The  property  of  a  double  refraction,  in  virtue  of  which  a  polarized  ray  is  unequally  divided  between  the  two 
images,  furnishes  us  with  a  most  convenient  and  useful  instrument  for  the  detection  of  polarization  in  a  beam 
of  light,  and  for  a  variety  of  optical  experiments.  It  is  nothing  more  than  a  prism  of  a  doubly  refracting 
medi'im  rendered  achromatic  by  one  of  glass,  or  still  better,  by  another  prism  of  the  same  medium  properly 
disposed,  so  as  to  increase  the  separation  of  the  two  pencils.  The  former  method  is  simple;  and,  when  large 
refracting  angles  are  not  wanted,  the  uncorrected  colour  in  one  of  the  images  is  so  small  as  not  to  be  trouble- 
some. It  is  most  convenient  to  make  the  refracting  angle  such  as  to  produce  an  angular  separation  of  about  2° 
between  the  images.  Thus,  in  fig.  174,  let  A  B  C  G  F  be  a  prism  of  Iceland  spar,  cut  in  such  a  manner  (we 
-  will  at  present  suppose)  that  the  refracting  edge  C  G  shall  contain  the  axis  of  the  crystal  ;  and  let  it  be  achro- 
matized as  much  as  possible  by  a  prism  of  glass  C  D  E  F  G.  Then,  if  Q  be  a  small,  colourless,  luminous  circle 
of  about  a  degree  or  two  in  apparent  diameter,  as  seen  by  an  eye  at  O,  the  interposition  of  the  combined  prisms 
will  divide  it  into  two,  Q  and  q.  Now,  if  the  light  of  Q  be  completely  unpolarized,  these  two  will  remain 
exactly  of  equal  intensity  while  the  prism  ABC  G  is  turned  round  in  a  plane  at  right  angles  to  the  line  of  vision. 
But  if  any  polarity  exist  in  the  original  light,  the  two  images  Q,  </  will,  in  turning  round  the  prism,  appear  alter- 
nately more  and  less  bright  one  than  the  other  ;  and  being  always  seen  immediately  side  by  side,  the  least 
inequality,  and  consequently  the  least  admixture  of  polarized  light  in  the  incident  beam,  will  be  detected. 

Iceland  spar,  from  its  very  great  double  refraction,  is  commonly  used  for  these  prisms  ;  but  it  is  so  soft,  and 
its  structure  so  lamellar,  as  to  be  difficult  to  polish,  and  still  more  so  to  preserve  polished.  We  have  found  quartz 
and  limpid  topaz  to  answer  extremely  well.  The  following  ingenious  mode  of  rendering  available  the  low  double 
refraclion  of  the  former,  due  to  Dr.  Wollaston,  is  here  eminently  useful.  Let  A  B  C  D  abed  and  E  FGHefgh 
(nS-  175)  be  two  halves  of  a  hexagonal  prism  of  quartz  (the  form  it  affects)  produced  by  a  section  parallel  to  two  of 
the  sides.  In  the  vertical  face  AD  da  draw  any  line  L  K  parallel  to  the  sides,  and  therefore  to  the  axis  of  the  prism, 
(which  is  also  that  of  double  refraction,)  and  join  C  L,  ck.  Then  a  plane  CL  kc  will  cut  off  a  prism  CLKdcD, 
having  L  k,  D  d,  or  C  c,  for  its  refracting  edges,  either  of  which  is  parallel  to  the  axis.  Again,  in  the  other  half 
of  the  prism  join  E/"  and  H  g,  and  cut  the  prism  by  a  plane  passing  through  these  lines  ;  then,  regarding  either 
portion  as  a  double  refracting  prism,  having  for  refracting  edges  the  lines  E  H,  fg,  these  will  have  the  axis  of 
double  refraction  perpendicular  to  their  refracting  edges  ;  and,  in  particular,  the  axis  will  lie  in  the  faces  H  E  eh, 
or  F  G  gf  at  right  angles  to  H  E  or  fg.  If,  then,  we  take  care  to  make  the  refracting  angle  C  L  D  of  the 
prism  C  L  K  d  c  D  equal  to  that  of  the  edge  H  E  of  the  prism  II  E  cfg  h  ;  and  if  we  make  these  two  prisms 
act  in  opposition  to  each  other,  placing  the  edge  II  E  opposite  to  D  d,  and  the  edge  h  e  opposite  to  K  L  ;  and 
having  thus  brought  the  two  surfaces  D  L  kd  and  II  E  eh  in  contact,  cement  them  together  with  mastic,  or 
Canada  balsam,  it  is  evident,  that  their  principal  sections  will  be  at  right  angles  to  each  other  ;  and  therefore 
only  two  images  will  be  formed,  the  whole  of  the  extraordinary  ray  of  the  one  prism  passing  into  the  ordinary 
image  of  the  other,  and  vice  versd.  Now,  to  see  how  this  acts  to  double  the  separation  of  the  images,  let  us 
conceive  m  n  to  be  a  luminous  line  viewed  through  one  of  the  prisms  with  its  edge  downwards  and  horizontal. 
It  will  be  separated  into  two  images,  e  and  o,  the  one  more  raised  than  the  other.  Suppose  the  ordinary  image 
to  be  most  refracted.  Then,  if  we  interpose  the  other  prism  with  its  edge  upwards,  both  these  images  will 
be  refracted  downwards  ;  but  the  ordinary  image  o,  which  was  before  moat  raised,  now  undergoing  extraordinary 
refraction,  is  least  depressed,  and  comes  into  the  position  o  e,  while  the  extraordinary  one  e,  which  was  before 
least  raised  is  now  most  depressed,  and  comes  into  the  situation  eo  ;  and  it  is  evident  that  (the  refracting  angles 
being  equal,  and  the  double  refraction  of  the  two  prisms  the  same)  the  line  o  e  will  fall  as  far  short  of  the  ori- 
ginal line  mil,  as  eo  surpasses  it,  viz.  by  a  quantity  equal  to  the  distance  between  the  two  first  images  o  and  e; 
so  that  the  distance  between  the  twice  refracted  images  is  double  that  of  those  which  have  undergone  only  one 
refraction.  We  have  found  this  combination  extremely  advantageous,  as  quartz  takes  a  very  perfect  polish,  and 
from  its  hardness  is  not  liable  to  injury  from  scratches. 

Crystals  which  have  no  double  refraction  may  be  regarded  as  limits  of  those  which  have,  or  as  crystals  in 
which  the  two  rays  are  propagated  with  equal  velocity,  and  therefore  undergo  no  bifurcation  ;  or,  in  other  words, 
in  which  the  images  formed  coincide.  In  this  case  we  should  expect  to  find  no  polarization  of  the  emergent 
"f?nt'  because  the  two  pencils,  being  polarized  at  right  angles  to  each  other,  form  together  a  single  ray  having 
the  characters  of  unpolarized  light.  This  is  verified  by  experiment.  The  light  transmitted  by  fluor  spar,  for 


Fen  IV 


LIGHT.  515 

instance,  exhibits  no  signs  of  polarization,  unless  so  far  as  the  ordinary  action  of  the  surface  goes.     We  are  aware    Part  IV. 
>  of  no  experiments  indicating  how  far  the  action  of  the  surfaces  of  feebly  double  refracting  crystals  may  modify  >— - x — „ 
their  polarizing  forces,  or  rather  their  effects  on  a  ray  which  has  penetrated  below  the  surface  ;    or,  in  other 
•words,  how  far  piles  of  crystallized  lamina?  may  have  an  analogous  or  different  action  from  those  of  uncrystallized. 
Dr.  Brewster,  indeed,  found  piles  of  mica  films  to  polarize  light  by  transmission,  like  glass  piles,  but  the  subject 
is  open  to  further  inquiry. 

§  VII.   Of  the  Colours  exhibited  by  Crystallized  Plates  when  exposed  to  Polarized  Light,  and  of  the  Polarized 

Rings  which  surround  their  Optic  Axes. 

This  splendid  department  of  Optics  is  entirely  of  modern  and,  indeed,  of  recent  origin.  The  first  account  of  the 
colours  of  crystallized  plates  was  communicated  by  M.  Arago  to  the  French  Institute  in  1811,  since  which  period, 
by  the  researches  of  himself,  Dr.  Brewster,  M.  Biot,  M.  Fresnel,  and,  latterly,  also  of  M.  Mitscherlich,  and  others, 
it  has  acquired  a  developement  placing  it  among  the  most  important  as  well  as  the  most  complete  and  systematic 
branches  of  optical  knowledge.  As  might  be  expected,  under  such  circumstances,  as  well  as  from  the  state  of 
political  relations,  and  the  consequent  limited  intercourse  between  Britain  and  the  Continent  at  the  period  men- 
tioned, an  immense  variety  of  results  could  not  but  be  obtained  independently,  and  simultaneously,  or  nearly 
simultaneously,  on  both  sides  of  the  channel.  To  the  lover  of  knowledge,  for  its  own  sake, — the  philosopher, 
in  the  strict  original  sense  of  the  word, — this  ought  to  be  matter  of  pure  congratulation  ;  but  to  such  as  are 
fond  of  discussing  rival  claims,  and  settling  points  of  scientific  precedence,  such  a  rapid  succession  of  interesting 
discoveries  must,  of  course,  afford  a  welcome  and  ample  supply  of  critical  points,  the  seeds  of  an  abundant 
harvest  of  dispute  and  recrimination.  Regarding,  as  we  do,  all  such  discussions,  when  carried  on  in  a  spirit  of 
rivalry  or  nationality,  as  utterly  derogatory  to  the  interests  and  dignity  of  science,  and  as  little  short,  indeed,  of 
sacrilegious  profanation  of  regions  which  we  have  always  been  accustomed  to  regard  only  as  a  delightful  and 
honourable  refuge  from  the  miserable  turmoils  and  contentions  of  interested  life,  we  shall  avoid  taking  any  part 
in  them ;  and,  taking  up  the  subject  (to  the  best  of  our  abilities  and  knowledge)  as  it  is,  and  avoiding,  as  far  as 
possible,  all  reference  to  misconceived  facts  and  over-hasty  generalizations,  which  in  this  as  in  all  other  depart- 
ments of  science,  have  not  failed  (like  mists  at  daybreak)  to  spread  a  temporary  obscurity  over  a  subject 
imperfectly  understood,  shall  make  it  our  aim  to  state,  in  as  condensed  a  form  as  is  consistent  with  distinctness, 
such  general  facts  and  laws  as  seem  well  enough  established  to  run  no  hazard  of  being  overset  by  further 
inquiry,  however  they  may  merge  hereafter  in  others  yet  more  general ; — a  consummation  devoutly  to  be 
wished. 

The  general  phenomenon  of  the  coloured  appearances  to  which  this  section  is  devoted,  may  be  most  readily    .  °°'- 
and  familiarly  shown  as  follows.     Place  a  polished  surface  of  considerable  extent  (such  as  a  smooth  mahogany  ,,,'g^j    f 
table,  or,  what  is  much  better,   a  pile  of  ten  or  a  dozen  large  panes  of  glass  laid  horizontally)  close  to  a  exhibiting 
large  open  window,  from  which  a  full  and  uninterrupted  view  of  the  sky  is  obtained ;  and  having  procured  a  the  colours 
plate  of  mica,  of  moderate  thickness,  (about  a  thirtieth  of  an  inch,  such  as  may  easily  be  obtained,  being  sold  of  crystal- 
in  considerable  quantity  for  the  manufacture  of  lanterns,)  hold  it  between  the  eye  and  the  table,  or  pile,  so  as  i'",danpc'^es' 
to  receive  and  transmit  the  light  reflected  from  the  latter  as  nearly  as  may  be  judged  at  the  polarizing  angle.  ;n  mjca 
In  this  situation  of  things,  nothing  remarkable  will  be  perceived,  however  the  plate  of  mica  be  inclined;  but  if 
instead  of  the  naked  eye  we  look  through  a  tourmaline  plate,  having  its  axis  vertical,  the  case  will  be  very  different. 
When  the  mica  plate  is  away,  the  tourmaline  will   destroy  the  reflected  beam,  and  the  surface  of  the  table,  or 
pile,  will  appear  dark  and  non-reflective  ;    at  least  in  one  point,  on  which  we  will  suppose  the  eye  to  be  kept 
steadfastly  fixed.     No  sooner  is  the  mica  interposed,  however,  than  the  reflective  power  of  the  surface  appears  to 
be   suddenly  restored  ;    and   on  inclining  the  mica  at  various   angles,  and  turning  it  about  in  its  own  plane, 
positions  will  readily  be  found  in  which  it  becomes  illuminated  with   the  most  vivid  and  magnificent  colours, 
which  shift  their  tints  at  the  least  change  of  position  of  the  mica,  passing  rapidly  from  the  most  gorgeous  reds 
to  the  richest  greens,  blues,  and  purples.     If  the  mica  plate  be  held  perpendicular  to  the  reflected  beam,  and 
turned  about  in  its  own  plane,  two  positions  will  be  found  in  which  all  colour  and  light  disappears  ;    and  the 
reflected  ray  is  extinguished,  as  if  no  mica  was  interposed.     Now,  if  we  draw  on  the  plate  with  a  steel  point  Two  re- 
two  lines  corresponding  to  the  intersection  of  the  mica  with  a  vertical  plane  passing  through  the  eye  in  either  markable 
of  these  two  positions,  we  shall  find  that  they  make  an  exact  right  angle.     For  the  moment,  let  us  call  these  lines  settions  of 
A  and  B  ;  and  let  a  plane  drawn  through  the  line  A,  perpendicular  to  the  plate,  be  called  the  section  A  ;   and  one  J-  6  c'Jjsta' 
similarly  drawn  through  the  line  B,  the  section  B.     Then  we  shall  observe  further,  that  when  we  turn  the  plate  '* 
irom  either  of  these  positions,  45°  round,  in  its  own  plane,  so  that  the   sections  A   and    B  shall   make   angles 
of  45°  with  the  plane  of  reflexion,  (i.  e.  of  polarization  of  the  incident  ray,)  the  transmitted  light  will   be  a 
maximum. 

If  the  thickness  of  the  mica  do  not  exceed  ^Oth  of  an  inch,  it  will  be  coloured  in  this  position ;  if  materially      886. 
greater,  colourless ;  and  if  less,  more  and  more  vividly  coloured,  and  with  tints  following  closely  the  succession  ^aw  of  .the 
of  the  reflected  series  of  the  colours  of  thin  plates,  and,  like  them,  rising  in  the  scale,  or  approaching  the  b'iu^at 
central  tint  (black)  as  the  thickness  is  less.     The  analogy  in  this  respect,  in  short,  is  complete,  with  the  excep-  perpendi- 
non  of  the  enormous  difference  of  thickness  between  the  mica  plate  producing  the  tints  in  question,  and  those  cular 
required  to  produce  the  Newtonian  rings.     It  appears  by  measures  made  in  the  manner  hereafter  to  be  described,  ' 
that   the  tint  exhibiteil  by  a  plate  of  mica  exposed  perpendicularly  to  the  reflected  ray,  as  above  described,  is 
the  same  with  that  reflected  by  a  plate  of  air  of  T^th  part  of  the  thickness  of  the  mica  employed. 

3x3 


510 


LIGHT. 


L'ght.  If  the  mica  (still  exposed  perpendicularly  to  the  ray)  be  turned  round  in  its  own  plane,  the  tint  does  not 
v-~"v™1'  change,  hut  only  diminishes  in  intensity  as  its  section  A  or  B  approaches  the  plane  of  polarization  of  the  inci-  ' 

887.  dent  light.  When,  however,  the  plate  is  not  exposed  perpendicularly,  this  invariability  no  longer  obtains;  and 
Tints  exhi-  {[je  changes  of  tint  appear  in  the  last  degree  capricious  and  irreducible  to  regular  laws.  In  two  situations, 
two  sections  however,  the  phenomena  admit  a  simple  view.  These  are  when  the  sections  A  and  B  are  both  45°  from  the 
tbo\e  plane  of  polarization,  and  the  mica  plate  is  inclined  backwards  and  forwards  in  the  plane  of  one  or  the  other  of 

mentioned,  these  sections.  This  condition  is  easily  attained  by  first  holding  the  plate  perpendicularly  to  the  reflected  ray  ; 
then  turning  it  in  its  own  plane  till  the  lines  A,  B  are  each  45°  inclined  to  the  vertical  plane,  then  finally  causing 
it  to  revolve  about  either  of  these  lines  as  an  axis.  It  will  then  be  seen  that  when  made  to  revolve  round  one  of 
them  (as  A)  or  in  the  plane  of  the  .section  B,  the  tint,  if  white,  will  continue  white  at  all  angles  of  inclination  ; 
but  if  coloured,  will  descend  in  the  scale  of  the  coloured  rings,  growing  continually  less  highly  coloured,  till  it  passes, 
after  more  or  fewer  alternations,  into  white;  after  which,  further  inclination  of  the  plate  will  produce  no  change. 
On  the  other  hand,  if  made  to  revolve  round  B,  or  in  the  plane  of  A,  the  tints  will  rise  in  the  scale  of  the  rings  ; 
and  when  the  mica  plate  is  inclined  either  way,  so  as  to  make  the  angle  of  incidence  about  35°  3',  will  have 
attained  its  maximum,  corresponding  to  the  black  spot  in  the  centre  of  Newton's  rings.  In  this  position  of  tin- 
plate,  the  reflected  beam  is  totally  extinguished  by  the  tourmaline,  as  if  the  sections  A  or  B  had  been  vertical. 
But  if  the  angle  of  incidence  be  still  further  increased  the  colours  reappear,  and  descend  again  in  the  scale  of 
the  rings,  passing  through  their  whole  series  to  final  whiteness.  We  take  no  notice  here  of  a  slight  deviation  from 
the  strict  succession  of  the  Newtonian  colours,  which  is  observed  in  the  higher  orders  of  the  tints,  as  we  shall 
have  more  to  say  respecting  it  hereafter. 

We  see,  then,  that  the  sections  A  and  B,  though  agreeing  in  their  characters  in  the  case  of  a  perpendicular 
exposure  of  the  mica,  yet  differ  entirely  in  the  phenomena  they  exhibit  at  oblique  incidences.  If  the  incidence 
take  place  in  the  plane  of  the  section  B,  the  tint  descends,  on  both  sides  of  the  perpendicular,  ad  infinitum. 
While,  if  the  incidence  be  in  the  section  A,  it  rises  to  the  central  black,  which  it  attains  at  equal  incidences  on 
either  side  of  the  perpendicular  (35°  3'),  and  then  descends  again  ad  infinitum,  or  to  the  composite  white  at  the 
other  extreme  of  the  scale. 

The  section  A,  then,  (which,  for  this  reason,  we  will  call  the  principal  section  of  the  mica  plate,)  is  characte- 
rised by  containing  two  remarkable  lines  inclined  at  equal  angles  to  the  surface  of  the  plate,  along  either  of 
which,  if  a  polarized  ray  be  incident,  its  polarization  will  not  be  disturbed  by  the  action  of  the  plate.  To  satisfy 
ourselves  of  this,  we  have  only  to  fix  the  mica  to  the  extremity  of  a  tube,  so  as  to  have  the  axis  of  the  tube 
inclined  at  an  angle  of  35°  3'  to  the  perpendicular  (or  54°  57'  to  the  plate)  in  the  plane  of  the  section  A  ;  then 
directing  the  axis  of  the  tube  to  the  centre  of  the  dark  spot,  or  the  reflecting  surface,  it  will  be  seen  to  continue 
optic  axes.  dark,  an(j  remain  so  while  the  tube  makes  a  complete  revolution  on  its  axis.  Now,  this  could  not  be  if  the 
*  mica  exercised  any  disturbing  power  on  the  plane  of  polarization.  Hence,  we  conclude,  that  the  two  lines  in 
question  possess  this  remarkable  property,  viz.  that  whatever  be  the  plane  of  polarization  of  a  ray  incident  along 
eitlicr  of  them,  it  remains  unaltered  after  transmission.  For,  although  in  the  experiment  above  described,  the 
plane  of  polarization  remained  fixed,  and  that  of  incidence  was  made  to  revolve,  it  is  obvious  that  the  reverse- 
process  would  come  to  the  very  same  thing. 

Now,  this  character  belongs  to  no  other  lines,  however  chosen,  with  respect  to  the  plate.  If  we  fix  the  plate 
on  the  end  of  the  tube  at  any  other  angle,  or  in  any  other  plane  with  respect  to  the  axis  of  the  latter,  although 
two  positions  in  the  rotation  of  the  tube  will  always  be  found  where  the  disappearance  of  the  transmitted  ray 
takes  place,  in  no  other  case  but  that  of  the  two  lines  in  question  will  this  disappearance  be  total,  or  nearly  so, 
in  all  points  of  its  revolution. 

The  refracting  index  of  mica  being  1.500,  an  angle  of  incidence  of  35°  3'  corresponds  to  one  of  refraction  = 
22°  31'.  Hence,  the  position  of  the  lines  within  the  mica  corresponding  to  these  external  lines  is  22J°  inclined 
to  the  perpendicular,  and  the  angle  included  between  them  45°.  These,  then,  are  axes  within  the  crystal, 
bearing  a  determinate  relation  to  its  molecules.  Dr.  Brewster  has  termed  them  axes  of  no  polarization,  a  long 
name.  M.  Fresnel,  and  others,  have  used  the  phrase  optic  axes,  to  which  we  shall  adhere.  As  this  term  has 
before  been  opplied  to  the  "  axes  of  no  double  refraction,"  we  must  anticipate  so  far  as  to  advertise  the  reader 
that  these,  and  the  "  axes  of  no  polarization,"  are  in  all  cases  identical. 

Having,  by  the  criteria  above  described,  determined  the  principal  section,  and  ascertained  the  situation  of 
the  optic  axes  of  the  mica  plate  under  examination,  let  the  plate  be  inclined  to  the  polarized  beam,  so  that  the 
rized  rings  jatter  shan  be  transmitted  along  the  optic  axes,  the  principal  section  A  making  an  angle  of  45°  with  the  plane  of 
optic' axes  polarization  ;  and  let  the  eye  (still  armed  with  the  tourmaline  plate,  with  its  axis  vertical)  be  applied  close  to 
General  de-  the  mica.  A  splendid  phenomenon  will  then  be  seen.  The  black  point  corresponding  to  the  direction  of  the 
scription  of  optic  axis  will  be  seen  to  be  surrounded  with  a  set  of  broad,  vivid,  coloured  rings,  of  an  elliptic,  or,  at  least,  oval 
their  pheno-  form,  divided  into  two  unequal  portions  by  a  black  band  somewhat  curved,  as  represented  in  fig.  176.  This 
band  passes  through  the  pole,  or  angular  situation  of  the  optic  axis,  about  which  the  rings  are  formed  as  a 
centre.  Its  convexity  is  turned  towards  the  direction  of  the  other  axis,  and  on  that  side  the  rings  are  also 
broader.  If,  now,  the  other  axis  be  brought  into  a  similar  position,  a  phenomenon  exactly  similar  will  be 
seen  surrounding  its  place,  as  a  pole.  If  the  mica  plate  be  very  thick,  these  two  systems  of  rings  appear  wholly 
detached  from,  and  independent  of,  each  other,  and  the  rings  themselves  are  narrow  and  close  ;  but  if  thin  (as  a 
30th  or  40th  of  an  inch)  the  individual  rings  are  much  broader,  and  especially  so  in  the  interval  between  the 
poles,  so  as  to  unite  and  run  together,  losing  altogether  their  elliptic  appearance,  and  dilating  towards  the  middle 
(or  in  the  direction  of  a  perpendicular  to  the  plate)  into  a  broad  coloured  space,  beyond  which  the  rings  are  no 
longer  formed  about  each  pole  separately,  but  assume  the  form  of  reentering  curves,  embracing  and  including 
both  poles.  Their  nature  will  presently  be  stated  more  at  large. 


Part  IV. 


888. 

Characters 
of  the  two 
most 

remarkable 
sections. 

889. 
The 

principal 
section 
defined. 
Contains 
the  two 


ol  these 
axes. 


890. 


891. 

Position  of 
the  optic 


892 
The  pola- 


mena. 
Fig.  176. 


LIGHT.  517 

Light.          If,  preserving  the  same  inclination  of  the  mica  plate  to  the  visual  ray,  it  be  turned  about  it  as  an  axis,  the   Part  IV. 
'  black  band  passing  through  the  pole  will  shift  its  place,  and  revolve  as  it  were  on  the  pole  as  a  centre  with  double  \— -^^-^^ 
the  angular  velocity,  so  as  to  obliterate  in  succession  every  part  of  the  ring's.     When  the   plate  has  made  45°       893. 
of  its  revolution,  so  as  to  bring  its  principal  section  into  the  plane  of  polarization  of  the  incident  beam,  this  Further 
band  also  coincides  in  direction  with  that  plane,  and  is  then  visibly  prolonged,  so  as  to  meet  that  belonging  to  Partl 
the  set  of  rings  about  the  other  pole  ;    and  is  crossed  at  the  middle  point  between  the  poles  by  another  dark 
space  perpendicular  to  it,  or  in  the  plane  of  the  section  B,  presenting  the  appearance  in  fig.  177.  Fig.  177. 

These  phenomena,  if  a  tourmaline  be  not  at  hand,  may  be  viewed,  (somewhat  less  commodiously,  unless  the      894. 
mica  plate  be  of  considerable  size,)  by  using  in  its  place  the  reflector  figured  in  fig.  170,  or  by  a  pile  of  glass  Other 
plates  interposed  obliquely  between  the  eye  and  the  mica.     In  this  manner  of  observing  them,  the  colours  are  jj^jy,^ 
surprisingly  vivid,  no  part  of  the  red  and  violet  rays  being  absorbed  more  than  the  rest;  whereas  the  tourmalines  these 'phe- 
generally  exert  a  considerable  absorbing  energy  on  these  rays  in  preference  to  the  rest,  and  thus  the  contrast  ofn0mena,. 
colours  is  materially  impaired.     On  the  other  hand,  however,  from  the  greater  homogeneity  of  the  transmitted 
light,  the  rings  are  more  numerous  and  better  defined  ;  and  in  this  respect  the  phenomenon  is  greatly  improved 
by  the  use  of  homogeneous  light. 

We  have  taken  mica  as  being  a  crystallized  body  very  easily  obtained  of  large  size,  and  presenting  its  axes 
readily,  and  without  the  necessity  of  artificial  sections.  It  is  thus  admirably  adapted  for  obtaining  a  general 
rough  view  of  the  phenomena,  preparatory  to  a  nicer  examination.  From  the  wide  interval  between  its  axes, 
however,  and  the  considerable  breadth  of  its  rings,  it  is  less  adapted,  when  employed  as  above  stated,  to  give  a 
clear  conception  of  the  complicated  changes  which  the  rings  undergo,  on  a  variation  of  circumstances.  For 
this  reason  we  shall  now  describe  another  and  much  more  commodious  mode  of  examining  the  systems  of 
polarized  rings  presented  by  crystals  in  general,  which  has  the  advantage  of  bringing  the  laws  of  their  pheno- 
mena so  evidently  under  our  eyes  as  to  make  their  investigation  almost  a  matter  of  inspection. 

It  is  evident,  that  when  we  apply  the  eye  close  to,  or  very  near  a  plate  of  mica,  or  other  body,  and  view,       896. 
beyond  it,  a  considerable  extent  of  illuminated  surface,  each  point  of  that  surface  will  be  seen  by  means  of  a  ray  General 
which  has  penetrated  the  plate  in  a  different  direction  with  respect  to  the  axes  of  its  molecules ;   so  that  we  may  principle  of 
consider  the   eye  as  in  the  centre  of  a  spherical  surface  from  all  points  of  which  rays  are  sent  to  it,  modified  ^w-" 
according  to  the  state  of  primitive  polarization,  and  the  influence  of  the  peculiar  energies  of  the  medium,  corre-  Hugs. 
spending  to  the  direction  in  which  they  traverse  it,  and  the  thickness  of  the  plate  in  that  direction. 

Any  means,  therefore,  by  which  we  can  admit  into  the  eye  through  the  plate  and  tourmaline  a  cone  of  rays  PeHscopic 
nearly  or  completely  polarized   in  one  general  direction,  or  according  to  any  regular  law,  will  afford  a  sight  of  tourmaline 
the  rings ;  and  therefore  exhibit,  at  a  single  view,  a  synopsis,  as  it  were,  of  the  modifications  impressed  on  an  aP 
infinite  number  of  rays  so  polarized  traversing  the  plate  in  all  directions.     The  property  of  the  tourmaline  so 
often  referred  to  puts  it  in  our  power  to  perform  this  in  a  very  elegant  and  convenient  manner,  by  the  aid  of  the 
little  apparatus  of  which  fig.  178  is  a  section.     ABCD  is  a  short  cylinder  of  brass  tube,  the  end  of  which,  AC,  Fig.  178. 
is  terminated  by  a  brass  plate,  having  an  aperture  a  b,  into  which  is  set  a  tourmaline  plate  cut  parallel  to  the 
axis:  hgik  is  another  similar  brass  cylinder,  provided  with  a  similar  aperture  and  a  similar  tourmaline  plate  G, 
and  fitted  into  the  former  so  as  to  allow  of  the  one  being  freely  turned  round  within  the  other  by  the  milled  edges 
B  D,  h  k.     A  lens  H  of  short  focus,  set  in  a  proper  cell,  is  screwed  on  in  front  of  the  tourmaline  G,  so  as  to 
have  its  focus  a  little  behind  its  posterior  surface,  (that  next  the  eye,  O.)     Between  the  two  surfaces  AC,  gi 
is  another  short  cylinder  of  thin  tube  c  d,  carrying  a  brass  plate  with  an  aperture  somewhat  narrower  than  those 
in  which  the  tourmalines  are  set,  and  on  which  any  crystallized  plate  F  to  be  examined  may  be  cemented  with 
a  little  wax.     This,  with  the  cylinder  to  which  it  is  fixed,  is  capable  of  being  turned  smoothly  round  within  the 
cylinder  ABCD  by  means  of  a  small  pin  e  passing  through  a  slit  f  made  in  the  side,  and  extended  round  so 
as  to  occupy  about  120°  of  the  circumference  ;  by  which  a  rotation  to  that  extent  may  be  communicated  to  the 
crystallized  plate  F  in  its  own  plane  between  the  tourmaline  plates.     The  pin  e  should  screw  into  the  ring  cd, 
that  it  may  be  easily  detached,  and  admit  the  ring  and  plate  to  be  taken  out  for  the  convenience  of  fixing  on  it 
other  crystals  at  pleasure. 

The  use  of  the  lens  H  is  to  disperse  the  incident  light,  and  thus  equalize  the  field  of  view  when  illuminated       897. 
by  any  source  of  light,  whether  natural  or  artificial,  as  well  as  to  prevent  external  objects  being  distinctly  seen  Mode  of 
through  it,  which  would  distract  the  attention  and  otherwise  interfere  with  the  phenomena.     The  rays  converged  a£f'on  ' 
by  the  lens  to  a  focus  within   the  crystallized  plate  F,  afterwards  diverge  and  fall  on  the  eye  O,  after  traversing  'alusappa~ 
the  plate  in  all  directions  within  the  limit  of  the  field  of  view.     As  by  this  contrivance   they  pass   through   a 
very  small  portion  of  the  crystal,  there  is  the  less  chance  of  accidental  irregularities  in  its  structure  disturbing 
the  regular  formation  of  the  rings,  since  we  have  it  in  our  power  to  select  the  most  uniform  portion  of  a  large 
crystal.     The  rays,  after  passing  through  the  lens,  are  all  polarized  by  the  tourmaline  G,  in  planes  parallel  to 
its  axis ;  and  passing  through  the  eye  in  this  state,  if  the  crystal  F  be  not  interposed,  the  rays  will,  or  will  not, 
penetrate  the  second  tourmaline,  according  as  its  axis  is  parallel  or  perpendicular  to  that  of  the  first.     In  con- 
sequence, when  the  cylinder  carrying  the  former  is  turned  round  within  that  carrying  the  latter,  the  field  of  view 
is  seen  alternately  bright  and  dark. 

When  the  crystallized  substance  F  is  interposed,  provided  it  be  so  disposed  that  one  or  other  of  its  optic  axes      fjgg 
is  situated  any  where  in   the  cone  of  rays  refracted  by  the  lens,  so  that  one  of  them  shall  reach  the  eye  by  Selection  of 
traversing  the  axis,  the  polarized  rings  are  seen.     If  both  the  axes  of  the  crystal  (supposing  it  to  have  more  crystals. 
than  one)  fall  within  the  field,  a  set  of  rings  will  be  seen  round  both,  and  may  be  studied  at  leisure.     In  order 
to  bring  the  whole  of  their  phenomena  distinctly  under  view,   it  is  requisite  to  select  such  crystals  as  have 
their  axes  not  much  inclined  to  each  other,  so  as  to  allow  the  rings  about  both  to  be  seen  without  the  necessity 
of  looking  very  obliquely  into  the  apparatus.     In  mica  the  axes  are  rather  tou  far  removed  for  this.     The  best 
crystal  we  can  select  for  the  purpose  is  nitre. 


51S  L  I  G  H  T. 

Light.         Nitre  crystallizes  in  long-,  six-sided  prisms,  whose  section,  perpendicular  to  tlieir  sides,  is  the  regular  hexagon.     1>jrt  IV. 
«"~v^  They  are  generally  very  much  interrupted  in  their  structure  ;    but  by  turning  over  a  considerable  quantity  of  <1-^v-»' 

899.  the  ordinary  saltpetre  of  the  shops,  specimens  are  readily  found  which  have  perfectly  transparent  portions  of 
'""J6'          some  extent.     Selecting  one  of  these,  cut  it  with  a  knife  into  a  plate  above  a  quarter  of  an  inch  thick,  directly 

preparing  across  the  axis  of  the  prism,  and  then  grind  it  down  on  a  broad,  wet  file,  till  it  is  reduced  to  about  ^th  or  ^th  inch 
and  polish-  m  thickness ;  smooth  the  surfaces  on  a  wet  piece  of  emeried  glass,  and  polish  them  on  a  piece  of  silk  strained 
ing  it.  very  tight  over  a  strip  of  plate  glass,  and  rubbed  with  a  mixture  of  tallow  and  colcothar  of  vitriol.  This  ope- 
ration requires  practice.  It  cannot  be  effected  unless  the  nitre  be  applied  wet,  and  rubbed  till  quite  dry, 
increasing  the  rapidity  of  the  friction  as  the  moisture  evaporates.  It  must  be  performed  in  gloves,  as  the  vapour 
from  the  fingers,  as  well  as  the  slightest  breatn,  dims  the  polished  surface  effectually.  With  these  precautions 
a  perfect  vitreous  polish  is  easily  obtained.  We  may  here  remark,  that  hardly  any  two  salts  can  be  polished 
by  the  same  process.  Thus,  Rochelle  salt  must  be  finished  wet  on  the  silk,  and  instantly  transferred  to  soft 
bibulous  linen,  and  rapidly  rubbed  dry.  Experience  alone  can  teach  these  peculiarities,  and  the  contrivances 
(sometimes  very  strange  ones)  it  is  necessary  to  resort  to  for  the  purpose  of  obtaining  good  polished  sections  of 
soft  crystals,  especially  of  those  easily  soluble  in  water. 

900.  The  nitre  thus  polished  on  both  its  surfaces  (which  should  be  brought  as  near  as  possible  to  exact  parallelism) 
Rings  ex-    [s  fo  be  placed  on  the  plate  at  F  ;    and  the  tourmaline  plates  being  then  brought  to  have  their  axes  at  right 

y     angles  to  each  other  (which  position  should  be  marked  by  an  index  line  on  the  cylinders)  1he  eye  applied  at  O, 

and  the  whole  held  up  to  a  clear  light,  a  double  system  of  interrupted  rings  of  the  utmost  neatness  and  beauty 

Fig.  179.     will  be  seen,  as  represented  in  fig.  179.     If  the  crystallized  plate  be  made  to  revolve  in  its  own  plane  between 

the  tourmalines  (which  both  remain  unmoved)  the  phenomena  pass  through  a  certain  series  of  changes  periodi- 

!r°  IH?      ca"y>  returning,  at  every  90°  of  rotation,  to  their  original  state.     Fig.  180  represents  their  appearance  when  the 

Fi'l  IS"*      rotation  is  just  commenced;  fig.  181,  when  the  angle  of  rotation  is  22J°,  or  67^°;   and  fig.  182,  when  it  equals 

45°.     When  the  tourmalines  are  also  made  to  revolve  on  each  other,  other  more  complicated  appearances  are 

produced,  of  which  more  presently.     We  shall  now,  however,  suppose  them   retained  in  the  situation  above 

mentioned,  i.e.  with  their  axes  crossed  at  right  angles,  and  proceed  to  study  the  following  particulars  : 

1.  The  form  and  situation  of  the  rings. 

2.  Their  magnitudes  in  the  same  and  different  plates. 

3.  Their  colours. 

4.  The  intensity  of  the  illumination  in  different  parts  of  their  periphery. 

901.  The  situation  of  the  rings  is  determined  by  the  position  of  the  principal  section  of  the  crystal,  or  by  that  of 
Situation  of  tne  Optic  axes  within  its  substance.     These  in  nitre   lie  in  a  plane  parallel  to  the  ax.is  of  the  prisms,  and  per- 

I1  pendicular  to  one  or  other  of  its  sides.  It  is  no  unusual  thing  to  find  crystals  of  this  salt  whose  transverse 
section  consists  of  distinct  portions,  in  which  the  principal  sections  make  angles  of  60°  with  each  other ;  indi- 
cating a  composite  or  macled  structure  in  the  crystal  itself.  These  portions  are  divided  from  each  other  by 
thin  films,  which  exhibit  the  most  singular  phenomena  by  internal  reflexion,  on  which  this  is  not  the  place  to 
enlarge.  In  an  uninterrupted  portion,  however,  the  forms  of  the  rings  are  as  represented  in  the  figures  above 
referred  to,  their  poles  subtending  at  the  eye  an  angle  of  about  8°.  Now,  it  is  to  be  remarked,  that  as  the  plate 
is  turned  round  between  the  tourmalines,  although  the  black  hyperbolic  curves  passing  through  the  poles  shift 
their  places  upon  the  coloured  lines,  and  in  succession  obliterate  every  part  of  them  ;  forming,  first,  the  black 
cross  in  fig.  179,  by  their  union  ;  then  breaking  up  and  separating  laterally,  as  in  fig.  180,  and  so  on.  Yet  the 
rings  themselves  retain  the  same  form  and  disposition  about  their  poles,  and,  except  in  point  of  intensity,  remain 
perfectly  unaltered  ;  their  whole  system  turning  uniformly  round  as  the  crystallized  plate  revolves,  so  as  to 
preserve  the  same  relations  to  the  axes  of  its  molecules.  Hence  we  conclude,  that  the  coloured  rings  are  related 
to  the  optic  axes  of  the  crystal,  according  to  laws  dependent  only  on  the  nature  of  the  crystal,  and  not  at  all  on 
external  circumstances,  such  as  the  plane  of  polarization  of  the  incident  light,  &c. 

902.  The    general   form  of  the  rings,  abstraction   made   of  the   black  cross,  is  as  represented    in    fig.   183.     If 
Form  of  the  we  regard  them   all  as    varieties    of  one  and  the  same   geometrical  curve,   arising  from   the   variation   of   a 
rings,           parameter  in  its  equation,  it  will  be  evident  that  this  equation  must,  in  its  most  general  form,  represent   a  re- 
entering  symmetrical  oval,  which  at  first  is  uniformly  concave,  and  surrounds  both  poles,  as  A  ;  then  flattens  at 

p™n'j^jtcs'  the  sides,  and  acquires  points  of  contrary  flexure,  as  B  ;  then  acquires  a  multiple  point,  as  C  ;  after  which  it 
breaks  into  two  conjugate  ovals  D  D,  each  surrounding  one  pole.  This  variation  of  form,  as  well  as  the  general 
figure  of  the  curves,  bears  a  perfect  resemblance  to  what  obtains  in  the  curve  well  known  to  geometers  under 
the  name  of  the  lemniscate,  whose  general  equation  is 

0s  4-  3/2  +  «2)a=  a*  (*2  +  4  •**)« 

when    the  parameter  b   gradually   diminishes  from  infinity  to  zero ;    2  a  representing    the  constant  distance 
between  the  poles. 

Q03  The  apparatus  just  described  affords  a  ready  and  very  accurate  method  of  comparing  the  real  form   of  the 

Verified  by  rings  with  this  or   any  other  proposed  hypothesis.     If  fixed  against  an  opening  in  the  shutter  of  a  darkened 

experiment,  room,  with  the  lens  H  outwards,  and  a  beam  of  solar  light  be  thrown  on  the  latter,  parallel  to  the  axis  of  the 

apparatus,  the  whole  system  of  rings  will  be  seen  finely  projected  against  a  screen  held  at  a  moderate  distance 

from  E.     Now,  if  this  screen  be  of  good  smooth  paper  tightly  stretched  on  a  frame,  the  outlines  of  the  several 

rings  may  easily  be  traced  with   a  pencil  on  it,  and  the  poles  being  in  like  manner  marked,  we  have  a  faithful 

representation  of  the  rings,  which  may  be  compared  at  leisure  with  a  system  of  lemniscates,  or  any  other  curve 

graphically  constructed,  so  as  to  pass  through  points  in  them  chosen  where  the  tint  is  most  decided.     This  has 


LIGHT.  519 

Light,     accordingly  been  done,  and  it  has  been  found  that  lemniscates  so  constructed  coincide  throughout  their  whole     Part  IV. 
"~V'~^  extent,  to  minute  precision,  with  the  outlines  of  the  rings  so  traced,  the  points  graphically  laid  down  falling  on  ' 
the  pencilled  outlines.     The  graphical  construction  of  these  curves  is  rendered  easy  by  the  well-known  property 
of  the  lemniscate,  in  which  the  rectangle  under  two  lines  PA  x  P'  A  drawn  from  the  poles  to  any  point  A  in 
the  periphery  is  invariable  throughout  the  whole  curve.     This  is  easily  shown  from  the  above  equation,  and  the 
value  of  this  constant  rectangle  in  any  one  curve  is  represented  by  a  x  b. 

When  we  shift  from  one  ring  to  another,  a  remains  the  same,  because  the  poles  are  the  same  for  all.     To      904. 
determine  the  variation  of  6,  let  the  rings  be  illuminated  with  homogeneous  light,  (or  viewed  through  a  red  Variation  of 
glass,)  and  outlined  by  projection,  as  above.     Then,  if  we  determine  the  actual  value  of  06  by  measuring  the  the  Pa™.~ 
lengths  of  two  lines  P  A,  P'  A  drawn  from  P,  P'  to  any  point  of  each  curve ;  and,  calculating  their  product,  (to  arithmetic 
which  a  b  is  equal,)  it  will  be  found  that  this  product,  and  therefore  the  parameter  6,  increases  in  the  arithmetical  progression 
progression  0,  1,  2,  3,  4,  Sfc.  for  the  several  dark  intervals  of  the  rings  beginning  at  the  pole,  and  in  the  progres-  f">m  ring  to 
sion  £,  -|,  4.  &c.  for  the  brightest  intermediate  spaces.     To  ensure  accuracy,  the  mean  of  a  number  of  values  of  ""£• 
PA  X  P'A,  at  different  points  of  the  periphery,  may  be  taken  to  obviate  the  effect  of  any  imperfection  in  the 
crystal. 

This,  then,  is  the  law  of  the  magnitudes  of  the  successive  rings  formed  by  one  and  the  same  plate.     But  if  we      905. 
determine  the  value  of  the  same  product  for  plates  of  nitre  similarly  cut,  but   of  different  thicknesses,    or  Effect  of 
of  the  same  reduced  in  thickness  by  grinding,  it  will  be  found  to  vary  inversely  as  the  thickness  of  the  plate,  vary'ng  'he 
ccnteris  paribus.  JJ*™88  of 

The  colours  of  the  polarized  rings  bear  a  great  analogy  to  those  reflected  by  thin  plates  of  air,  and  in  most     &gr\R 
crystals  would  be  precisely  similar  to  them  but  for  a  cause  presently  to  be  noticed.     In  the   situation  of  the  j^e  coiours 
tourmaline  plates  here  supposed  (crossed  at  right  angles)  they  are  those  of  the  reflected  rings,  beginning  with  a  Of  the  rings. 
black  centre,  at  the  pole.      If  examined  in  the  situation  of  fig.  179,  and  traced  in   a  line  from  either  pole 
cutting  across  the  whole  system,  at  right  angles  to  the  line  joining  the  poles,  they  will  almost  precisely  follow 
the  Newtonian  scale  of  tints.     For  the  present  we  will  suppose  that  they  do  so  in  all  directions.     It  is  evident, 
then,  that  each  particular  tint  (as  the  bright  green  of  the  third  order,  for  instance)  will  be  disposed  in  the  form 
of  a  lemniscate,  and  will  have  its  own  particular  value  of  the  product  a  b.     The  tint,  then,  may  be  said  to  be 
corresponding  to, — dependent  on, — or,  if  we  will,  measured  by  a  b.     In  conformity  with   this  language  the  Numerical 
coloured  curves  have  been  termed,  and  not  inaptly,  isochromatic  lines.     Now,  in  the  colours  of  thin  plates,  we  measure  of 
have  seen  that  these  tints  arise  from  a  law  of  periodicity  to  which  each  homogeneous  ray  is  subject ;  and  that  ^  '',"' 
(without  entering  at  this  moment  into  the  cause  of  such  periods)  the  successive  maxima  and  minima  of  each  par-  ^(^  ijne 
ticular  coloured  ray  passed  through,  in  the  scale  of  tints,  correspond  to  successive  multiples  by  ^,  |>  -|,  A,  &c.  of 
the  period  peculiar  to  that  colour.     In  the  colours  of  thin  plates,  the  quantity  which  determines  the  number  of 
periods  is  the  thickness   of   the  plate  of  air,  or  other  medium  traversed  ;    and  the  number  of  times  a  certain 
standard  thickness  peculiar  to  each  ray  is  contained  therein,  determines  the  number  of  periods,  or  parts  of  a 
period,  passed  through.     In  the  colours,  and  in  the  case  now  under  consideration,  the  number  of  periods  is  Lawof  pe- 
proportional  to  the  product  (0  x  #')  of  the  distances  from  either  pole,  for  one  and  the  same  thickness  of  plate, —  nodieSty. 
and  for  different  plates  to  t  the  thickness, — and,  therefore,  generally,  to  0  X  Q1  X  t,  provided  we  neglect  the 
effect  of  the  inclination  of  the  ray  in  increasing  the  length  of  the  path  of  the  rays  within  the  crystal,  or  regard 
the  whole  system  of  rings  as  confined  within  very  narrow  limits  of  incidence. 

This  condition  obtains  in  the  case  here  considered,  because  of  the  proximity  of  the  axes  in  nitre  to  each  other      907. 
and  to  the  perpendicular  to  the  surfaces  of  the  plate.     But  in  crystals  such  as  mica,  or  others  where  they  are  Transition 
still  wider  asunder,  it  is  not  so ;  and  the  projection  of  the  isochromatic  curves  on  a  plane  surface  will   deviate  'rom  mtre 
materially  from  their  true  form,  which  ought  to  be  regarded  as  delineated  on  a  sphere  having  the  eye,  or  rather  a  c<L°ta[sr 
point  within  the  crystal,  for  a  centre.     In  such  a  case,  it  might  be  expected  that  the  usual  transition  from  the  whose  axes 
arc  to  its  sine  should  take  place  ;  and  that,  instead  of  supposing  the  tint,  or  value  of  a  b,  to  be  proportional  are  farther 
simply  to  0  x  0'  x  t,  (putting  0  =  A  P,  and  0'  =  AP',)  we  ought  to  have  it  proportional  to  sin  0  x  sin  ff  x  asunder- 
length  of  the  path  of  the  ray  within  the   crystal.     Now  (putting  p  for  the  angle  of  refraction,  and  t  for  the 
thickness  of  the  plate)  we  have  t  .  sec  p  —  length  of  the  ray's  path  within  the  crystal.     If,  then,  we  put  n  for 

the  number  of  periods  corresponding  to  the  tint  ab  for  the  ray  in  question,  and  suppose  h  =  ,  or  the 

n 

unit  whose  multiples  determine  the  order  of  the  rings,  we  shall  have 

a  b               t  General  ex- 

n  =  =   -r-  .  sin  0  .  sin  0' .  sec  p,                (a)                                         pression  for 

h                h  the  tint 

.  polarized 

and  h  = .  sin  0  .  sin  0'.  (6)  by  any 

n  .  COS  p  crystallized 

plate. 

If,  then,  the  suppositions  made  be  correct,  we  ought  to  have  the  function  on  the  right  hand  side  of  this  last 
equation  invariable,  in  whatever  direction  the  ray  penetrates  the  crystallized  plate,  and  whatever  be  the  order  of 
the  tint  denoted  by  n.  We  shall  here  relate  only  one  experiment,  to  show  how  very  precisely  the  agreement  of 
this  conclusion  with  fact  is  sustained. 

A  ray  of  light  was  polarized  by  reflexion  at  a  plate  of  perfectly  plane  glass,  and  transmitted  through  a  plate      908. 
of  mica,  having  its  principal  section  45°  inclined  to  the  plane  of  primitive  polarization,  and  the  mica  plate  Experiment 
made  to  revolve  in  the  plane  of  its  principal  section  about  an  axis  at  right  angles  thereto,  (or  about  the  axis  B,  verifying 
Art.  885.)     In  this  state  of  things,  if  viewed  through  a  tourmaline  as  above  described,  or  by  other  more  refined '  ' 


520 


LIGHT. 


means  presently  to  be  noticed,  the  succession  of  tints  exhibited  by  the  mica  was  that  of  a  section  of  the  rings 
in  fig.  182,  made  by  a  line  drawn  through  both  the  poles.  To  render  the  observation  definite,  a  red  glass  was 
interposed  so  as  to  reduce  the  rings  to  a  succession  of  red  and  black  bands,  and  the  angles  of  incidence  corre- 
sponding to  the  maxima  and  minima  of  the  several  rings  very  accurately  measured.  These  are  set  down  in 
Col.  2  of  the  following  table.  Col.  1  contains  the  values  of  n,  0  corresponding  to  the  pole,  £  to  the  first 
maximum,  1  to  the  first  minimum,  1^  to  the  second  maximum,  and  so  on.  The  third  column  contains  the 
angles  of  refraction  computed  for  an  index  1.500  ;  the  fourth  and  fifth,  those  of  0  and  &  ;  the  sixth,  those  of  A 
deduced  from  the  above  equation,  and  which  ought  to  be  constant.  The  excesses  above  the  mean  are  stated 
in  the  last  column,  and  show  how  very  closely  that  equation  represents  the  fact.  The  thickness  of  the  mica  was 
0.023078  inches  =  t. 


Values  of  ». 

An^le*  of  in- 
cidence. 

Angles  of 
refraction  =:  f. 

Values  of  t. 

Values  of  6'  '. 

Values  of  ft. 

Excesses  above 
the  mean. 

0.0 

35°  3'  30" 

22°31'    0" 

0°    0'     0" 

45°   21    0" 

0.5 

32  5  3  20 

21   14  40 

1    16  20 

43  45  40 

0.032952 

-  0.000195 

1.0 

30  34  40 

19  49  30 

2  41  30 

42  20  30 

0.033622 

-f  0.000475 

1.5 

28  15  40 

18  24     0 

470 

40  55     0 

0.033035 

-  0.000112 

2.0 

25  34  20 

16  43  30 

5  47  30 

39   14  30 

0.033327 

-f  0.000180 

2.5 

22  46  20 

14   57   15 

7  33  45 

37  28   15 

0.03314S 

-f-  0.000001 

3.0 

19  35  40 

12  55   10 

9  35  50 

35  26  10 

0.033058 

-  0.000089 

3.5 

15  48  40 

10  27   50 

12     3   10 

32  58  50 

0.033026 

-  0.000121 

4.0 

10  48  50 

7   11    10 

15   19   53 

29  42  10 

0.033010 

-  0000137 

909.  Proceeding  thus,  and  measuring  across  the  system  of  rings  in  all  directions  for  plates  of  various  crystals  and 

Gener.il  of  all  thicknesses,  it  has  been  ascertained,  as  a  general  fact,  that  in  all  substances  which  possess  the  property  of 
establish-  developing  periodical  colours  by  exposure  to  polarized  light  in  the  manner  described,  the  tint  («),  or  rather 
ment  of  the  the  number  of  periods  and  parts  of  a  period  corresponding,  in  the  case  of  a  ray  of  given  refrangibility,  to  a 

thickness  t,  an  angle  of  refraction  p,  and  a  position  within  the  crystal,  making  angles  0  and  0'  with  the  optic 

axes,  is  represented  by  the  equation 


law. 


t 


sec  p          .     .  . 
—  X  sin  0  .  sm  &, 


Case  of  a 

crystal 
formed  into 
a  sphere. 


910. 

Methods  of 
viewing  the 
rings  at 
great  obli- 
quities. 


Fig.  184. 


911. 

Rings  in 

crv-t.il'i 
with  one 
axis. 
Fig.  185. 


A  being  a  constant  depending  only  on  the  nature  of  the  crystal  and  the  ray.  Were  the  crystal  of  a  spherical  form, 
instead  of  a  parallel  plate,  t.  sec  />,  which  represents  the  path  traversed  by  the  ray  within  it,  must  be  replaced 
by  a  constant  equal  to  the  diameter  of  the  sphere,  and  in  that  case  the  tint  would  be  simply  proportional  to 
the  product  of  the  sines  of  0  and  0'.  This  elegant  law  is  due  to  M.  Biot,  though  it  is  to  Dr.  Brewster's  inde- 
fatigable and  widely  extended  research  that  we  owe  the  general  developement  of  the  splendid  phenomena  of  the 
polarized  rings  in  biaxal  crystals.  It  appears,  then,  from  this,  that  if,  on  the  surface  of  a  sphere  formed  of  any 
crystal,  curves  analogous  to  the  lemniscate,  or  having  sin  0  x  sin  ff  constant  for  each  curve,  and  varying  in 
arithmetical  progression  from  curve  to  curve,  be  described, — then,  if  the  sphere  be  turned  about  its  centre  in  a 
polarized  beam,  as  above  described,  the  tint  polarized  at  every  point  of  each  curve  will  be  the  same,  and  in 
passing  from  curve  to  curve  will  obey  the  law  of  periodicity  proper  to  the  crystal. 

There  is  hardly  any  character  in  which  crystals  differ  more  widely  than  in  the  angular  separation  of  their  optic 
axes,  as  the  table  annexed  to  the  end  of  this  article  will  show.  This,  while  it  affords  most  valuable  criteria  to 
the  chemist  and  mineralogist,  in  discriminating  substances  and  pointing  out  differences  of  structure  and  com- 
position which  would  otherwise  have  passed  unnoticed,  renders  the  investigation  of  their  phenomena  difficult, 
since  it  is  frequently  impossible,  by  any  contrivance,  to  bring  both  the  axes  under  view  at  once  ;  and  neces- 
sitates a  variety  of  artifices  to  obtain  a  sight  of  the  rings  about  both.  It  is  often  very  easy  to  cut  and  polish 
crystallized  bodies  in  some  directions,  and  very  difficult  in  others.  However,  by  immersing  plates  of  them  in 
oil,  and  turning  them  round  on  different  axes,  or  by  cementing  on  their  opposite  sides  prisms  of  equal  refracting 
angles  oppositely  placed,  as  in  fig.  184,  we  may  look  through  them  at  much  greater  obliquities  than  without  such 
aid ;  and  thus,  by  increasing  the  range  of  vision  to  nearly  a  hemisphere,  avoid  in  most  instances  the  necessity 
of  cutting  them  in  different  directions. 

When  the  two  axes  coalesce,  or  the  crystal  becomes  uniaxal,  the  lemniscates  become  circles;  and  the  block 
hyperbolic  lines,  passing  through  the  poles,  resolve  themselves  into  straight  lines  at  right  angles  to  each  other, 
forming  a  black  cross  passing  through  the  centre  of  the  rings,  as  in  fig.  185.  In  this  case  the  tint  is  repre- 
sented by  t  .  sin  0* ;  and  in  the  case  of  plates,  where  t,  the  thickness,  is  considerable,  or  where,  from  the  other- 
wise peculiar  nature  of  the  substance  the  rings  are  of  small  dimensions,  0  is  small,  and  therefore  proportional 
to  its  sine;  so  that  in  passing  from  ring  to  ring  Cft  increases  in  arithmetical  progression.  Hence  the  diameters 
of  the  rings  are  as  the  square  roots  of  the  numbers  0,  1,  2,  3,  ftc.;  and  therefore  their  system  is  similar,  with 
the  exception  of  the  black  cross,  to  the  rings  seen  between  object-glasses.  Carbonate  of  lime  cut  into  a  plate 
at  right  angles  to  the  axis  of  its  primitive  rhomboid,  exhibits  this  phenomenon  with  the  utmost  beauty.  The 
most  familiar  instance,  however,  may  be  found  in  a  sheet  of  clear  ice  about  an  inch  thick  frozen  in  still  weather. 
A  pane  of  window-glass,  or  a  polished  table  to  polarize  the  light,  a  sheet  of  ice  freshly  taken  up  in  winter 


LIGHT.  521 

produce  the  rings,  and  a  broken  fragment  of  plate  glass  to  place  near  the  eye  as  a  reflector,  arc  all  the  apparatus     Part  IV. 
required  to  produce  one  of  the  most  splendid  of  optical  exhibitions.  v— v— «^ 

If  0  be  not  very  small,  the  measure  of  the  tint,  instead  of  t  .  sin  6*,  is  t  .  seep  .  sin  6*.  We  have  seen  that  in  912. 
uniaxal  crystals,  s'in  0a  is  proportional  to  the  difference  of  the  squares  of  the  velocities  v  and  v'  of  the  ordinary  Analogy 
and  extraordinary  ray,  or  to  v"*  —  v*.  Now,  if  we  denote  by  ^  and  T*  the  times  taken  by  these  two  rays  to  "^'"J^ 

traverse  the  plate,  we  have  v  =  —    —  and  v'  =  -          —  ;    therefore  t  .  sec  .  p  sin  0*  is  proportional  to  rizedPrin»* 

\  i    _L    '•»  (         >\  and  thos' 

(t  .  sec  #.*(*£).  that  is,  to  (-  ±-^f       >  .  «  sec  ,)»,  tfg*** 


\  interference 

or  (which  is  the  same  thing)  to  (v  +  v')  .  v  v1  (T  —  T').  But,  neglecting  the  squares  of  very  small  quantities,  of 
the  order  v'  —  v  and  T  —  T',  for  such  they  are  in  the  immediate  neighbourhood  of  the  axis,  the  factors  v  -j-  v'  and 
v  v  are  constant ;  so  that  the  tint  is  simply  proportional  to  T  —  T',  the  difference  of  times  occupied  by  the  two 
rays  in  traversing  the  plate  ;  or  the  interval  of  retardation  of  the  slower  ray  on  the  quicker.  This  very  remark- 
able analogy  between  the  tints  in  question  and  those  arising  from  the  law  of  interferences,  was  first  perceived 
by  Dr.  Young ;  and,  assisted  by  a  property  of  polarized  light  soon  to  be  mentioned,  discovered  by  Messrs. 
Arago  and  Fresnel,  leads  to  a  simple  and  beautiful  explanation  of  all  the  phenomena  which  form  the  subject  of 
this  section,  and  of  which  more  in  its  proper  place. 

The  forms  of  the  rings  are  such  as  we  have  described,  only  in  regular  and  perfect  crystals;  every  thing  which       913 
disturbs  this  regularity,  distorts  their  form.     Some  crystals  are  very  liable  to  such  disturbances,  either  arising  Circum- 
from  an  imperfect  state  of  equilibrium,  or  a  state  of  strain  in  which  the  molecules   are  retained,  or  to  actual  w?'.lcfs 
interruptions  in  their  structure.     Thus,  specimens  of  quartz  and  beryl  are  occasionally  met  with,  in  which  the  jjs^rt  tj,e 
single  axis  usually  seen  is  very  distinctly  separated  into  two,  the  rings  instead  of  circles  have  oval  forms,  and  the  rinKS, 
black  cross  (which  in  cases  of  a  well  developed  single  axis  remains  quite  unchanged  during  the  rotation  of  the  cry- 
stallized plate  in  its  own  plane)  breaks  into  curves  convex  towards  each  other,  but  almost  in  contact  at  their  vertices, 
at  every  quarter  revolution.     Cases  of  interruption  occur  in  carbonate  of  lime  very  commonly,  and  in  muriacite 
perpetually ;   and  the  effects  produced  by  them  on  the  configurations  of  the  rings  rank  among  the  most  curious 
and  beautiful  of  optical  phenomena.     They  have  not,  however,  been  anywhere  described,  and  our  limits  will 
not  allow  us  to  make  this  article  a  vehicle  for  their  description. 

The  form  of  the  rings  being,  then,  considered,  let  us  next  inquire  more  minutely  into  their  colours.     These       914. 
being  all  composite,  and  arising  from  the  superposition  on  each  other  of  systems  of  rings  formed  by  each  homo-  Colours  of 
geneous  ray,  we  can  obtain  a  knowledge  of  their  constitution  only  by  examining  the  rings  in  homogeneous  the  ra>s- 
light.     This  is  easy,  for  we  have  only  to  illuminate  the  apparatus  described  above  by  homogeneous  light  of  all 
degrees  of  refrangibility  from  red  to  violet,  by  passing  a  prismatic  spectrum  from  one  end  to  the  other  over  the 
illuminating  lens  H,  the  eye  being  applied  as  usual  at  O,  and  observe  the  changes  which  take  place  in  the  rings, 
in  passing  from   one  coloured  illumination  to  another;    and,  if  necessary,  measure  their  dimensions.     This  is 
readily  done,  either  by  projecting  them  on  a  screen  in  a  darkened  room,  as  described  in  Art.  903,  or  by  detaching 
the  lens  H,  fig.  178,  and  simply  looking  through  the  apparatus  at  a  sheet  of  white  paper  strongly  illuminated 
by  the  rays  of  a  prismatic  spectrum,  where  the  rings  will  appear  as  if  depicted  on  the  paper,  and  their  outlines 
easily  marked,  or  their  diameters  measured.     The  following  are  the  general  facts  which  may  thus  be  readily 
verified. 

First,  in  the  case  of  crystals  with  a  single  axis,  the  rings  remain  circular,  and  their  centres  are  coincident  for       915. 
all  the  coloured  rays,  but  their  dimensions  vary.     In  the  generality  of  such  crystals,  their  diameters  for  different  "n.  c|7sla1' 
refrangibilities  follow  nearly  the  law  of  the  Newtonian   rings,  when   viewed  in   similar  illuminations  ;   their  Wl 
squares  (or  rather  the  squares  of  their  sines)  being  proportional,  or  nearly  so,  to  the  lengths  of  the  fits,  or  of  the  Deviations 
undulations  of  the  rays  forming  them.     This  law,  however,  is  very  far  from  universal  ;   and  in  certain  crystals  is  from  New- 
altogether  subverted.     Thus,  in  the  most  common  variety  of  apophyllite,  (from  Cipit,  in  the  Tyrol, — not  from  ton's  scale 
Fassa,  as  is  commonly  stated,)  the  diameters  of  the  rings  are  nearly  alike  for  all  colours,  those  of  the  green  rings  '"  'j1?  aP°" 
being  a  very  little  less ;   those  formed  by  rays  at  the  confines  of  the  blue  and  indigo  ex€btly  equal,  and  those  p  1J  Ile' 
of  violet  rays  a  little  greater  than  the  red  rings.     It  is  obvious,  that  were  the  rings  of  all  colours  exactly  equal, 
the  system  resulting  from  their  superposition  would  be  simple  alternations  of  perfect  black  and  white,  continued 
ad  infiniliim.     In  the  case  in  question,  so  near  an  approach  to  equality  subsists,  that  the  rings  in  a  tourmaline 
apparatus  appear  merely  black  and  white,  and  are  extremely  numerous,  no  less  than   thirty-five  having  been 
counted,  and  many  of  those  too  close  for  counting  being  visible  in  a  thick  specimen. 

When  examined  more  delicately,  colours  are,  however,  distinguished,  and  are   in  perfect  conformity  with  the       gjg 
law  stated,  being  for  the  first  four  orders  as  follow  : 

First  order.     Black,  greenish  white,  bright  white,  purplish  white,  sombre  violet  blue. 

Second  order.  Violet  almost  black,  pale  yellow  green,  greenish  white,  white,  purplish  white,  obscure  indigo 

inclining  to  purple. 

Third  order.     Sombre  violet,  tolerable  yellow  green,  yellowisn  white,  white,  pale  purple,  sombre  indigo. 
Fourth  order.    Sombre  violet,  livid  grey,  yellow  green,  pale  yellowish  white,  white,  purple,  very  sombre 
indigo,  &c. 

Carbonate  of  lime,  beryl,  ice,  and  tourmaline  (when  limpid)  are  instances  of  uniaxal  crystals,  in  whose  rings       917 
the  Newtonian  scale  of  tints  is  almost  exactly  imitated ;  and,  consequently,  the  intervals  of  retardation   of  the 
ordinary  and  extraordinary  rays  of  any  colour  on  one  another,  are  proportional  to  the  lengths  of  their  undu- 
lations.    On  the  other  hand,  in  the  hyposulphate  of  lime,  we  are  furnished  with  an  instance  of  more  rapid 

VOL.  iv.  3  Y 


522  L  I  G  H  T. 

Light,      degradation  of  tints,  and  therefore  of  a  more  rapid  variation  of  the  interval  just  mentioned.    The  following  was    1'ari  iv 
the  scale  of  colour  of  the  rings  observed  in  this  remarkable  crystal  :  v—  -  v~~* 

iac"          First  or^er'     B'ack,  very  faint  sky  blue,  pretty  strong  sky  blue,  very  light  bluish   white,  white,  yellowish 

white,  bright  straw  colour,  yellow,  orange  yellow,  fine  pink,  sombre  pink. 
Second  order.  Purple,  blue,  bright  greenish  blue,  splendid  green,  light  green,  greenish  white,  ruddy  white, 

pink,  fine  rose  red. 

Third  order.     Dull  purple,  pale  blue,  green  blue,  white,  pink. 
Fourth  order.    Very  pale  purple,  very  light  blue,  white,  almost  imperceptible  pink. 
After  which  the  succession  of  colours  was  no  longer  distinguishable. 

918.  A  degradation  still  more  rapid  has  been  observed  in  certain  rare  varieties  of  uniaxal  apophyllite,  accompanied 
Other  re-     with  remarkable  and  instructive  phenomena.     In  these,  the  diameters  of  the  rings  (instead  of  diminishing  as  the 
markable      refrangibility  of  the  light  of  which  they  are  formed  increases)  increase  with  great  rapidity,  and  actually  become 
deviation      iifini'6  f°r  rays  of  intermediate  refrangibility  ;   after  which  they  again  become  finite,  and  continue  to  contract 

up  to  the  violet  end  of  the  spectrum,  where,  however,  they  are  still  considerably  larger  than  in  the  red  rays.  In 
consequence  of  this  singularity,  their  colours  when  illuminated  with  white  light  furnish  examples  of  a  complete 
inversion  of  Newton's  scale  of  tints.  The  following  were  the  tints  exhibited  by  two  varieties  of  the  mineral  in 
question,  in  one  of  which  the  critical  point  where  the  rings  become  infinite  took  place  in  the  indigo,  and  in  the 
other  in  the  yellow  rays.  In  the  former  they  were 

First  order.     Black,  sombre  red,  orange,  yellow,  green,  greenish  blue,  sombre  and  dirty  blue. 

Second  order.  Dull  purple,  pink,  ruddy  pink,  pink  yellow,  pale  yellow   (almost  white,)  bluish  green,  dull 
pale  blue. 

Third  order.     Very  dilute  purple,  pale  pink,  white,  very  pale  blue. 
In  the  latter  variety,  the  tints  were 

First  and  only  order.     Black,  sombre  indigo,  indigo  inclining  to  purple,  pale  lilac  purple,  \-ery  pale  reddish 
purple,  pale  rose  red,  white,  white  with  a  hardly  perceptible  tinge  of  green. 

919.  The  doubly  refracting  energy  of  a  crystal  may  be  not  improperly  measured  by  the  difference  of  the  squares 
Relation       of   the  velocities  of   an  ordinary  and  extraordinary  ray  similarly  situated  with  respect  to  the   axes  ;    but  as 
between  the  tn|s  difference,  for  rays  variously  situated  in  one  and  the  same  crystal,  is  proportional  to  sin  ff2,  or  in  biaxal 

rihe'rinos  crys'a's  to  sm  ^  •  sm  &'-  *ne  intrinsic  double  refractive  energy  of  any  crystal  may  be  represented  by 
and  the  C*  —  v'* 

doubly  e  =  -.  —    —  .—z;  (c) 

refractive  sin  6  •  SIn  °  ?  _      ,^ 

regarding  this  henceforth  as  the  definition  of  this  energy,  we  have,  in  uniaxal  crystals,  e  =  --  —  —  ,  and 
this  will  evidently  measure  the  actual  amount  of  separation  of  two  such  rays  when  emergent  from  the  crystal. 
If  in  this  we  put  for  v  and  v'  their  equals  —  '•  -  and  -  —  -j  -  ,  we  shall  have,  after  reduction, 


In  a  parallel  plate,   perpendicular  to  the  axis  and  in  the  immediate  vicinity  of   the  axis,  v'  and  sec  p  may 

be  regarded  as  constant,  and  c2  —  v1*  is  proportional   to  t'  —  T,  the  interval  of  retardation  of  one  ray  on  the 

other,  to  which  the  tint  in  white  light  and  the  number  of  periods  and  parts  of  a  period  in  homogeneous  light 

(to  which,  for  brevity,  we  will  continue  to  extend  the  term  tint)  are  proportional.     We  see,  then,  that  in  such 

cases  the  intrinsic  double  refracting  energy  is  directly  as  the  tint  polarized,  and  inversely  as  sin  &*,  and  therefore 

also  inversely  as  the  squares  of  the  diameters  of  the  rings.     As  the  rings  increase  in  magnitude,  then,  ceeteris 

paribus,  the  double  refractive,  energy  diminishes  ;   and  hence  a  very  curious  consequence  follows,  viz.  that  in  the 

two  cases  last  mentioned  it  vanishes  altogether  for  those  colours  where  the  rings  are  infinite  ;   in  other  words, 

that  although  the  crystal  be  doubly  refractive  for  all  the  other   coloured  rays,  there  is  one  particular  ray  in  the 

Case  of        spectrum  (viz.  the  indigo  in  the  former,  and  the  yellow  in  the  latter  case)  with  respect  to  which  its  refraction  is 

crystals  at    s|ngie      Jn  the  passage  through  infinity,  there  is  generally  a  change  of  sign.     In  the  instances  in  question  this 

tractive"       change  takes  place  in  the  value  of  e  or  v*  —  v",  which  passes  from  negative  to  positive.     And  the  spheroid  of 

repulsive,     double  refraction  changes  its  character  accordingly  from  oblate  to  prolate,  passing  through  the  sphere  as  its 

and  neutral,  intermediate  state.     The  manner  in  which  this  may  be  recognised,  without  actually  measuring,  or  even  perceiving 

its  double  refraction,  will  be  explained  further  on. 

920.          For  crystals  with  two  axes  we  have  only,  at  present,  the  ground  of  analogy  to  go  upon  in  applying  the 

Application  above  formula  and  phraseology  to  their  phenomena.     The  general  fact  of  an  intimate  connection  of  the  double 

to  biaxal      refracting  energy  with  the  dimensions  of  the  rings,  is  indeed  easily  made  out  ;  for  it  is  a  fact  easily  verified  by 

crystals.       experiment,  that  all  crystals,  whether  with  owe  or  two  axes,  in  which  the  rings  or  lemniscates  formed  are  of 

small   magnitude  in  respect  of  the  thickness  of  the   plate   producing  them,  arc  powerfully  double  refractive, 

and  vice  versd  ;  and  that,  generally  speaking,  the  separation  of  the  ordinary  and  extraordinary  pencils  is,  ceeterii 

paribus,  greater  in  proportion  as  the  rings  are  more  close  and  crowded  round  their  poles.     In  uniaxal  crystals, 

in  which  Jhe  laws  of  double  refraction  are  comparatively  simple,  there  is  little  difficulty  in  submitting  the  point 

to  the  test  of  direct  experiment  and  exact  measurement,  and  it  is  found  to  be  completely  verified.     In  biaxal, 

however,  such  precise  and  direct  comparison  is  more  difficult,  and  calls  for  a  knowledge  of  the  general  laws  of 

double  refraction.     The  analogy,  however,  supported  by  the  general  coincidence  above  mentioned,  is  too  strong 

to  be  refused  ;  and,  as  we  advance,  will  be  found  to  gain  strength  with  every  step. 


LIGHT. 


523 


Light.          In  biaxal  crystals,  similar  deviations  from  exact  proportionality  between  the   lengths  of  the  periods  of  the    Part  IV. 
^-y—  ^  several  coloured  rays  and  those  of  their  undulations,  or  fits,  exist  ;    but  their  effect  in  disturbing  the  colours  of  —  —  ~,,-~-^ 
the  rings  is  interfered  with,  and  frequently  masked  by,  another  cause,  which  has  no  existence  in  uniaxal  crystals,      921. 
viz.  that  the   optic  axes  differ  in  situation,  within  one  and  the  same  crystal  for  the    differently  refrangible  Separation 
homogeneous  rays;   and,  therefore,  that  the  elementary  lemniscates,  whose  superposition  forms  the  composite  of  tlie  "I1!.": 
fringes  seen  in  a  white  illumination,  differ  not  only  in  magnitude  but  in  the  places  of  'their  poles  and  the  interval  "rentl   '  ' 
between  them.     To  make  this  evident  to  ocular  inspection,  take  a  crystal  of  Rochelle  salt,  (tartrate  of  soda  and  refrangible 
potash,)  and  having  cut  it  into  a  plate  perpendicular  to  one  of  its  optic  axes,  or  nearly  so,  and  placed  it  in  a  rays  in 
tourmaline  apparatus,  let  the  lens  H  be  illuminated  with  the  rays  of  a  prismatic  spectrum,  in  succession,  begin-  h'axa' 
ning  with  the  red  and  passing  gradually  to  the  violet.     The  eye  being  all  the  time  fixed  on  the  rings,  they  will  crJstals- 
appear  for  each  colour  of  perfect  regularity  of  form,  remarkably  well  defined,  and  contracting  rapidly  in  size  as 
the  illumination  is  made  with  more  refrangible  light  ;  but  in  addition  to  this,  it  will  be  observed,  that  the 
whole  system  appears  to   shift  its  place  bodily,  and  advance  regularly  in  one  direction   as  the   illumination 
changes  ;  and  if  it  be  alternately  altered  from  red  to  violet,  and  back  again,  the  pole,  with  the  rings  about  it, 
will  also  move  backwards  and  forwards,  vibrating,  as  it  were,  over  a  considerable  space.     If  homogeneous  rays 
of  two  colours  be  thrown  at  once  on  the  lens,  two  sets  of  rings  will  be  seen,  having  their  centres  more  or  less 
distant,  and  their  magnitudes  more  or  less  different,  according  to  the  difference  of  refrangibility  of  the  two  species 
of  light  employed. 

Since  the  plate  in  this  experiment  is  supposed  to  have  its  surfaces  perpendicular  to  the  mean  position  of  the       922. 
optic  axis,  the  cause  of  these  appearances  cannot  be  found  in  a  mere  apparent  displacement  of  the  rings  by  A"  tlle  a*es 
refraction  at  the  surface,  existing  to  a  greater  extent  for  the  violet  than  the  red  rays,  add  to  which,  that  the  angle  ''?  in  ''le, 
which  their  poles  describe,  is  neither  the  same  in  magnitude  nor  direction  for  different  crystals.     In  some,  the  principal  "" 
optic  axes  approach  each  other  in  violet  light,  and  recede  in  red  ;  while  in  others  the  reverse  is  the  case.     In  all,  section. 
however,  so  far  as  we  are  aware,  the  optic  axes  for  all  the  coloured  rays  lie  in  one  plane,  viz.  the  principal 
section  of  the  crystal.     This  is  rendered  matter  of  inspection  by  cutting  any  crystal  so  that  both  axes  shall  be 
visible  in  the  same  plate,  and  placing  it  with  its  principal  section  in  the  plane  of  primitive  polarization.     In  this 
state  of  things  the  first  ring  about  each  pole,  as  in  fig.  179,  is  seen  divided  into  two  halves,  and  puts  on,  if 
the  plate  be  pretty  thick,  the  appearance  of  two  semi-elliptic  spots,  one  on  each  side  of  the  principal  section. 
These  spots  are  observed  to  be  differently  coloured  at  their  two  extremities  •  m  some  crystals  the  ends  of  the 
spots,  as  well  as  the  segments   of  the  rings  adjacent  to  them,  which  are  turned  towards  each  other,  being 
coloured  red,  and  the  other,  or  more  distant  ends,  with  blue  ;  and  in  others,  the  reverse.     In  some  crystals  this 
coloration  is  slight,  and  in  a  very  few,  imperceptible  ;  but  in  others  it  is  so  great,  that  the  spots  are  drawn 
out  into  long  spectra,  or  tails  of  red,  green,  and  violet  light  ;  and  the  ends  of  the  rings  are  in  like  manner 
distorted   and  highly  coloured,  presenting  the  appearance  in  fig.  186.     This  is  the  case  with  Rochelle  salt,  pig.  186. 
above  mentioned.     If  these  spectra  be  examined  with  coloured  glasses,  or  with  homogeneous  light,  they  will  be 
seen  to  be  composed  as  in  fig.  187,  by  the  superposition  of  well  defined  spots  of  the  several  simple  colours  pjg  137 
arranged  in  lines  on  each  side  of  the  principal   section.     In  the  case  of  Rochelle  salt,  the  angular  extent  of 
these  spectra,  within  the  medium,  which  measures  the  interval  between  the  optic  axes  for  violet  and  red  rays, 
amounts  to  no  less  than  10°. 

Dr.  Brewster  has  given  the  following  list  of  crystals  presenting  these  phenomena,  which  he  has  divided  into       ^23. 
two  classes,  according  to  his  peculiar  and  ingenious  views.  Dr' 


Class  I. 

Nitre. 

Sulphate  of  baryta. 

Sulphate  of  strontia. 

Phosphate  of  soda. 

Tartrate  of  potash  and  soda. 

Supertartrate  of  potash  and  soda. 

Arragonite. 

Carbonate  of  lead.  (?) 

Sulphato-carbonate  of  lead. 

ClassII. 
Topaz. 

Mica. 
Anhydrite. 
Native  borax. 
Sulphate  of  magnesia. 


Unclassed. 
Chromate  of  lead. 
Muriate  of  mercury. 
Muriate  of  copper. 
Oxynitrate  of  silver. 
Sugar. 

Crystallized  Cheltenham  salts. 
Nitrate  of  mercury. 
Nitrate  of  zinc. 
Nitrate  of  lime. 
Superoxalate  of  potash. 
Oxalic  acid. 
Sulphate  of  iron. 
Carbonate  of  lead.  (?) 
Cymophane. 
Felspar 
Benzoic  acid. 
Chromic  acid. 
Nadelstein  (Faroe.) 


viations  of 
tint  fronl 


To  which  list  a  great  many  more  might  be  added.     Bicarbonate  of  ammonia,  indeed,  is   the  only  biaxal  crystal 

we  have  examined  in  which  the  optic  axes  for  all  colours  appear  to  be  strictly  coincident.  924 

This  separation  of  the  axes  of  different  colours  explains  a  remarkable   appearance  presented  by  the  rings  of  Phenomena 
all  biaxal  crystals,  when  placed  with  their  principal  section  45°  inclined  to  the  plane  of  polarization  of  the  incident  of  the  vir- 
light.     It  is  universally  observed  that,  in  traversing  the  whole  system  of  rings  in  the  plane  of  the  principal  tual  Poles 

3^2  explained, 


524  LIGHT. 

Light,  section,  the  nearest  approximation  to  Newton's  scale  of  colours  is  obtained  by  assuming1,  for  the  origin  of  the  P»rt  IV. 
VN^~V"^"''  scale,  not  the  poles  themselves,  but  other  points  (which  have  been  called  virtual  poles,  though  improperly)  lying  *— •-~«— «• 
either  between  or  beyond  them,  according  to  the  crystal  examined,  and  at  a  distance  from  them,  inva- 
riable for  each  species  of  crystal,  whatever  he  the  thickness  of  the  plate.  In  consequence,  the  poles 
themselves  are  not  absolutely  black,  but  tinged  with  colour  ;  and  their  tint  descends  in  the  scale  as  the  thickness 
of  the  plate  increases,  and  as,  in  consequence,  one,  two,  or  more  orders  of  rings  intervene  between  them  and  the 
points  from  which  the  scale  originates.  These  points  are  observed  to  lie  between  the  poles  in  all  crystals  which 
have  the  blue  axes  nearer  than  the  red,  such  as  Rochelle  salt,  borax,  mica,  sulphate  of  magnesia,  topaz;  and 
beyond  them  for  those  in  which  the  red  axes  include  a  less  angle  than  the  blue,  as  sulphate  of  baryta,  nitre, 
arragonite,  sugar,  hyposulphite  of  strontia ;  and  this  fact,  as  well  as  the  constancy  of  their  distance  from  the 
poles  when  the  thickness  of  the  plate  is  varied,  renders  their  origin  evident.  In  fact,  since  the  violet  rings  are 
smaller  than  the  red,  if  the  centre  about  which  the  former  are  described,  instead  of  being  coincident  with  that 
of  the  latter,  be  shifted  in  either  direction,  carrying  its  rings  with  it,  someone  of  the  violet  rings  will  necessarily 
be  brought  up  to,  and  fall  upon  a  red  ring  of  the  same  order ;  and  the  same  holding  good  with  the  intermediate 
rays,  provided  the  law  which  determines  the  separation  of  the  different  coloured  axes  be  not  very  different  from 
that  which  regulates  the  dimensions  of  the  rings  of  corresponding  colours,  the  point  of  coincidence  of  a  red 
and  violet  ring  of  the  same  order  will  be  nearly  that  of  a  red  and  green,  or  any  intermediate  colour.  The  tint, 
then,  at  this  point  will  be  either  absolutely  black,  (if  they  be  dark  rings  which  are  thus  brought  to  coincidence,) 
or  white,  if  bright ;  and  from  this  point  the  tints  will  reckon  either  way  with  more  or  less  exactness,  accord- 
ing to  the  same  scale  which  would  have  held  good  had  the  points  of  coincidence  been  the  poles  themselves. 
Should,  however,  the  two  laws  above  mentioned  differ  very  widely,  an  uncorrected  colour  will  be  left  at  the 
point  of  nearest  compensation,  just  as  happens  when  two  prisms  whose  scales  of  dispersion  are  dissimilar  are 
employed  to  achromatise  each  other.  To  what  an  extent  the  disturbance  of  the  Newtonian  scale  of  tints  may 
be  carried  by  this  and  the  other  causes  already  explained,  the  reader  may  see  by  turning  to  the  table  of  tints 
exhibited  by  Rochelle  salt  inPM.  Trans.  1820,  part  i. 

925.  We  come  next  to  consider  the  law  of  the  intensity  of  the  illumination  of  the  rings  in  different  parts  of  their 
T»-o  suppo-  periphery  ;  but  this  part  of  their  theory  will  require  us  to  enter  more  fundamentally  into  the  mode  in  which  their 
sitions  as  to  formation  is  effected,  and  to  examine  what  modifications  the  polarized  ray  incident  on  the  crystallized   plate 

0  undergoes  in  its  passage  through  it,  so  as  to  present  phenomena  so  totally  different  from  those  which  it  would 
crystals  in  have  offered  without  such  intervention.  It  is  evident  then,  first,  that  since  the  ray,  if  not  acted  on  by  the  plate, 
forming  the  would  have  been  entirely  stopped  by  the  second  tourmaline,  but,  when  so  acted  on,  is  partially  transmitted  so  as 
rings.  to  exhibit  coloured  appearances  of  certain  regular  forms ;  that  the  crystallized  plate  must  have  either  destroyed 

altogether  the  polarization  of  that  part  of  the  light  which  is  thereby  enabled  to  penetrate  the  second  tourmaline, 
or,  if  not,  must  have  altered  its  plane  of  polarization,  so  as  to  allow  of  a  partial  transmission.     Between  these 
Doctrine  of  two  suppositions  it  is  not  difficult  to  decide.     Were  the  portion  of  light  which  passes  through  the  second  tourma- 
inilarization  ]jne  ancj  forms  tne  rings  wholly  depolarized,  that  is,  restored  to  its  original   state   of  natural  light,   since   the 
re  '  remainder,  its  complement  to  unity,  which  continues  to  be  stopped  by  the  tourmaline,  retains  its  state  of  polariza- 
tion unaltered,  it  is  evident,  that  each  ray  at  leaving  the  crystallized  plate  would  be  composed  of  two  portions, 
one  unpolarized  (=  A),  the  other  (=  1  —  A)  polarized.     Of  these,  the  half  only  of  the  first  (^  A)  would  be 
transmitted   by  the  second   tourmaline.     Now,  suppose  this  to  be  turned  round  in  its  own  plane  through  any 
angle  (=  a)  from  its  original  position,  then  the  unpolarized  portion  will  continue  to  be  half  transmitted;    and 
the   polarized,  being   now  partially  also    transmitted,  (in  the  ratio  of  sin*  a  :  1,)  will   mix   with  it,  so  that  the 
compound  beam  will  be  represented  by 

£  A  +  ( 1  —  A)  .  sin8  a  =  sin*  a  -f-  —  .  cos  2  a. 

Now,  if  we  suppose  a  to  pass  in  succession  through  the  values  0,  45°,  90°,  135°,  180°,  &c.,  this  will  become 
respectively  £  A,  £,  1  —  £  A,  J,  £  A,  &c.  Hence,  at  every  quarter  revolution  the  tints  ought  to  change  from 
those  of  the  reflected  rings  to  those  of  the  transmitted,  the  complements  of  the  former  to  white  light;  and  at 
every  half  quarter  revolution  no  rings  at  all  should  be  seen,  but  merely  an  uniformly  bright  field  illuminated 
with  half  the  intensity  of  light  which  would  be  seen  were  the  second  tourmaline  altogether  removed. 

926.  But  the  phenomena  which  actually  take  place  are  very  different.     At  the  alternate  quadrants,  it  is  true,  the 
Phenomena  complementary  rings  are  produced,  and  the  appearance  is  as  represented  in  fig.  188.     The  black  cross  is  seen 

i  the  com-  changed  into  a  white  one ;  the  dark  parts  of  the  rings  are  become  the  bright  ones  ;  the  green  is  changed  into 
tary  red,  and  the  red  into  green,  &c. ;   so  that  if  we  were  to  examine  no  farther,  the  fact  would  appear  to  agree  with 
Fig.  188.      the  hypothesis.     But  in  the  intermediate  half  quadrants,  this  agreement  no  longer  subsists.     Instead  of  a  uni- 
formly illuminated  field,  a  compound  set  of  rings,  consisting  of  eight  compartments,  alternately  occupied  by 
the  primary  and  complementary  set,  is  seen,  presenting  the  appearance  of  fig.  191,  and  which  is  further  described 
in  Art.  935. 

927.  The  phenomena  then  are  incompatible  with  the  idea  of  depolarization.    It  remains  to  examine  what  account  can 
Hypothesis   be  given  of  them  on  the  supposition  of  a  change  of  polarization  operated   by  the  plate  ;  and  here  we  must 
of  a  change  remark  in  limine,  that  this  cause  is  what  in  Newton's  language  would  be  termed  a  vera  causa,  a  cause  actually 
ticn*0''     l"  *n  ex'stence;  f°r  we  nave  already  seen  that  every  ray,  whether  polarized  or  not,  traversing  a  double  refracting 

medium  in  any  direction,  except  precisely  along  its  axis,  is  resolved  into  two,  polarized  in  opposite  planes.  When 
the  incident  ray  is  polarized,  these  portions  (generally  speaking)  differ  in  intensity,  and  though,  owing  to  the 
parallelism  of  the  plate  they  emerge  superposed,  their  polarization  is  not  the  less  real,  and  either  of  them  may  be 
suppressed,  and  the  other  suffered  to  pass,  by  receiving  them  on  a  tourmaline  properly  situated.  This  is  so  far 


LIGHT.  525 

Light,      agreeable  to  the  observed  fact,  when  the  tourmaline  plate  next  the  eye  is  removed,  the  rays  of  which  the  two  sets    Part  IV. 
— ^~ •*>  of  rings  consist,  coexist  in  the  transmitted  cone  of  rays  whose  apex  is  the  eye,  but,  being  complementary  to  each  v— -v— » s 
other,  produce  whiteness.     This  may  be  made  matter  of  ocular  demonstration,  by  employing,  instead   of  a  !?°l|i sets  of 
tourmaline,  which  absorbs  one  image,  a  doubly  refracting  achromatic  prism,  of  sufficiently  large  refracting  angle  ™ 
to  separate  the  two  pencils  by  an  angle  greater  than  the  apparent  diameter  of  the  system  of  rings,  when  the" 
primary  set  will  appear  in  one  image,  and  its  complementary  set  in  the  other  ;  meanwhile,  to  return  to  our  tour- 
malines, since  the  two  sets  of  rings  seen  in  the  two  positions  of  the  posterior  tourmaline  are  complementary,  it 
follows,  that  all  the  rays  suppressed  in  one  position  are  transmitted  in  that  at  right  angles  to  it,  and  vice  versa  ; 
and,  as  a  necessary  consequence,  that  every  pair  of  corresponding  rays  in  the  primary  and  complementary  set  are 
polarized  in  opposite  planes. 

The  only  thing,  then,  which  appears  mysterious  in  the  phenomena  thus  conceived,  is  the  production  of  colour.      92S. 
A  doubly  .refracting  crystal,  which  receives  a  polarized  ray  of  whatever  colour,  divides  it  between  its  two  pencils,  M'  Blot's 
according  to  a  ratio  dependent  only  on  the  situation  of  the  planes  of  polarization  and  of  incidence,  and  of  the  ^° 
axes  of  the  crystal,  and  not  at  all  on  its  refrangibility.     How  then  happens  it,  that  at  certain  angles  of  incidence  polarization. 
the  red  rays  pass  wholly  into  one  image,  and  the  green  or  violet  into   the  other,  while  at  other  incidences  the 
reverse  takes  place  :    whence,  in  short,  arises  the  law  of  periodicity  observed.     To  answer  this  question,  M.  Biot 
imagined  his  theory  of  alternate,  or  as  he  terms  it  movable  polarization,  according  to  which,  as  soon  as  a  pola- 
rized ray  enters  into  a  thin  crystallized  lamina,  its  plane  of  polarization  commences  a  series  of  oscillations,  or 
rather  alternate  assumptions  per  saltum  of  two  different  positions,  one  in  its  original  plane,  the  other  in  a  plane 
making  with  that  plane  double  the  angle  which  the  principal  section  of  the  crystal  makes  with   it.     These 
alternations  he  supposes  to  be  more  frequent  for  the  more  refrangible  rays,  and  to  recur  periodically,  like  New- 
ton's fits  of  easy  reflexion  and  transmission,  at  equal  intervals  all  the  time  the  ray  is  traversing  the  crystal,  which 
intervals  are  shorter  the  more  inclined  its  path  is  to  the  axis  or  axes.     This  theory  is  remarkably  ingenious  in 
its  details;  and  in  its  application  to  the  phenomena  of  the  rings,  though  open  (as  stated  by  its  author)  to  certain 
obvious  criticisms,  is  yet,  we  conceive,  capable  of  being  regarded  as  a  faithful  representation  of  most  of  their 
leading  features.     There  is,  however,  one  objection  against  it  of  too  formidable  a  nature  to  allow  of  its  being  Objection 
received  unless  explained  away,  if  any  other  can  be  devised  not  open  to  the  same  or  greater.     It  is,  that  it  requires  against  "• 
us  to  consider  the  action  of  a  thin  crystal  on  light  as  totally  different,  not  merely  in  degree,  but  in  kind,  from 
that  of  a  thick  one,  while  yet  it  marks  no  limit  by  which  we  are  to  determine  where  its  action  as  a  thin  crystal 
ceases,  and  that  proper  to  a  thick  one  commences,  nor  establishes  any  gradations  by  which  one  mode  of  action 
passes  into  the  other.     A  thick  crystal,  as  we  know,  polarizes  the  rays  ultimately  emergent  from  it  in  two  planes, 
dependent  only  on  the  position  of  the  crystal  and  that  of  the  ray,  while  M.  Biot's  theory  makes  the  position  of 
the  plane  of  polarization  of  the  incident  ray  an  element  in  determining  their  ultimate  polarization  by  a  thin  one. 
Nor  are  we  in  this  theory  to  regard  as  thin  crystals  only  films  or  delicate  laminae.     A  plate  of  a  tenth  of  an  inch 
thick  or  more  may  be  a  thin  plate  in  some  cases  of  feebly  polarizing  bodies,  such  as  apophyllite,  &c. 

As  the  apparatus  employed  by  M.  Biot  for  studying  the  phenomena  of  the  colours  of  thin  crystallized  plates  929. 
offers  great  conveniences  for  the  measurement  of  the  angles  at  which  different  tints  are  produced,  and  for  their  51.  Biot's 
exhibition  in  their  state  of  greatest  purity  and  contrast,  we  shall  here  describe  it,  and  state  some  of  the  chief  oeneral 
results  at  which  he  has  arrived.  A  (fig.  189)  is  a  plane  glass  blackened  at  the  posterior  surface,  or  a  plate  of  ^5*^6 ed 
obsidian  inclined  at  the  polarizing  angle  to  the  axis  of  a  tube  A  B,  so  as  to  reflect  along  it  a  polarized  ray ;  (if  fig.  igg, 
greater  intensity  be  required,  we  may  use  a  pile  of  glass  plates,  taking  care  that  they  be  of  truly  parallel  surfaces,  190. 
and  placed  exactly  parallel  to  each  other.)  B  C  is  a  tube,  stiffly  movable  round  A  B  as  an  axis,  having  a 
graduated  ring  at  B,  read  off  by  a  vernier  attached  to  the  tube  A  B,  and  carrying  two  arms,  G  and  H,  through 
which  the  axis  of  a  swing  frame  E  passes,  which  can  thus  be  inclined  at  any  angle  to  the  common  axis  of  the 
tubes,  its  inclination,  or  the  angle  of  incidence  of  the  ray  reflected  along  the  axis  on  the  plane  of  the  frame 
being  read  oft'  by  an  index  on  the  divided  lateral  circle  D.  In  this  frame  is  an  aperture  F,  in  which  turns  a 
circular  plate  of  brass  having  a  hole  in  its  centre,  over  which  is  fastened  with  wax  the  crystallized  plate  to  be 
examined,  and  which  can  thus  be  turned  round  in  its  own  plane,  independently  of  any  motion  of  the  rest  of  the 
apparatus,  so  as  to  place  its  principal  section  in  any  azimuth  with  respect  to  the  plane  of  incidence.  We  have 
found  it  convenient  to  have  this  part  of  the  apparatus  constructed  as  in  fig.  190,  where  a  is  the  square  plate  of 
the  frame  ;  b  a  divided  circle  movable  in  it  and  read  off  by  an  index  ;  e,  d  is  a  circular  plate  movable  within 
the  divided  circle  to  admit  of  adjustment,  after  which  it  is  fastened  in  its  place  by  a  little  clamp,  so  as  to  turn 
with  the  circle  ;  this  carries  in  its  centre  another  swinging  circle  e,  moving  stiffly  on  its  axis,  and  having  in  the 
middle  an  aperture,  over  which  the  crystal  is  cemented,  thus  giving  room  for  an  adjustment  of  the  plane  of  the 
surface  of  incidence,  in  case  it  be  not  exactly  at  right  angles  to  the  principal  section  of  the  crystal,  an  adjustment 
very  useful  when  artificial  surfaces  are  under  examination,  which  it  is  hardly  possible  to  cut  and  polish  with 
perfect  precision.  It  is  also  convenient  for  some  experiments  to  have  a  second  frame  similar  to  the  first,  placed 
on  the  prolongation  of  the  arms  G,  H.  M  is  a  doubly  refracting  prism,  rendered  achromatic  either  by  a  prism 
of  flint  glass,  or,  still  better,  by  another  prism  of  the  same  doubly  refracting  medium.  Two  prisms  of  quartz, 
arranged  as  in  Art.  882,  are  very  convenient.  Their  angles  should  be  such,  that  when  placed  at  M  the  two 
images  of  a  small  aperture  P,  in  a  diaphragm  near  the  end  of  the  tube,  should  appear  almost  in  contact.  The 
prisms  so  adjusted  are  mounted  on  a  stand  N,  independent  of  the  other  apparatus,  and  capable  of  being  turned 
round  by  an  arm  K,  carrying  a  vernier,  by  whose  aid  the  angle  of  rotation,  or  position  of  the  plane  in  which  the 
double  refraction  takes  place,  can  be  read  off  on  a  divided  circle  L.  The  prism  should  be  so  adjusted  in  its  cell, 
that  when  the  vernier  reads  off  zero,  the  extraordinary  image  should  be  extinguished  ;  and  when  90°,  the  ordinary. 
Occasionally  a  tourmaline  plate  or  a  glass  reflector  may  be  substituted  for  the  prism. 

To  use  this  apparatus,  the  crystallized  lamina  (which  we  will  at  present  suppose  to  be  a  parallel  plate  of  any      930. 


526  LIGHT. 

Light,     uniaxal   crystal,  having  its  axis  perpendicular  to  the  plane  of  the  plate,)  is  to   be  placed   on  the  swing  frame    Parl  IV- 
>— v— -'  across  the  aperture,  and  being  adjusted  so  as  to  have  its  axis  directed  precisely  along  the  axis  of  the  tube  when  V— ~v~~~ 
th»  the  vernier  of  I)  reads  off  zero,  which  is  readily  performed  by  the  various  adjustments  belonging  to  the  frame, 
tus-    as  above  described,  the  instrument  is  ready  for  use.     The  attainment  of  this  condition  may  be  known  by  turning 
the  tube  C  on  the  tube  A  B  as  an  axis,  when  the  extraordinary  image  of  the  aperture  P,  seen  through  a  doubly 
refracting  prism,  ought  to  vanish  in  the  zero  position  of  the  vernier  K,  and  not  be  restored  in  any  part  of  the 
rotation  of  the  tube;  for  it  is  manifest,  that  the  axis  is  the  only  line  to  which  this  property  belongs,  or  to  which 
all  the  rings  are  symmetrical.     It  is  then  evident,  that,  however  the  parts  of  the  apparatus  be  disposed,  1st,  the 
reading  off  of  the  vernier  D  will  give  the  angle  of  incidence  on  the  plate  ;  2d,  that  of  the  vernier  B,  the  angle 
made  by  the  plane  of  incidence  with  the  plane  of  primitive  polarization  ;  3d,  that  of  the  vernier  c  will  indicate 
the  angle  included  by  any  assumed  section  of  the  crystallized  plate  perpendicular  to  its  plane  with  the  plane  of 
incidence  ;  and,  lastly,  that  the  reading  of  the  vernier  K  will  give  the  angle  between  the  plane  of  primitive  polari- 
zation and  the  principal  section  of  the  doubly  refracting  prism. 

Suppose  now  we  adjust  the  vernier  B   to  zero,  it  will  then  be  found,  that  however  the  plate  E  be  situated,  or 
Ti^f'th*"  whatever  be  the  incidence  of  the  ray,  only  the  ordinary  image  will  be  seen  (being  white,)  the  extraordinary  being 
phenomena  ext'ng'u'shed  (or  black.)     In  this  case  we  traverse  the  system  of  rings  in  the  direction  of  the  vertical  arm  of  the 
of  the  rings  black  cross,  fig.  185,  of  the  primary,  and  the  white  one  of  the  complementary  set,  see  fig.  188.     The  phenomena 
of  one  axis,  are  the  same  if  we  set  the  vernier  B  to  90°,  and  then  turn  the  frame  E  on  its  axis,  thus  varying  the  incidence  in 
Fig.  188.     a  plane  at  right  angles  to  that  of  primitive  polarization,  or,  which  comes  to  the  same  thing,  traversing  the  rings 
along  the  horizontal  arm  of  the  black  and  white  crosses.     In  intermediate  positions  of  the  vernier  B,  we  traverse 
the  rings  along  a  diameter,  making  an  angle  with  vertical  equal  to  the  reading  of  the  vernier.     In  this  case  the 
two  images  of  P  are  both  visible,  and  finely  coloured  ;  the  extraordinary  image  presenting  the  tint  of  the  primary 
rings  due  to  the  particular  angle  of  incidence  indicated  by  the  vernier  D  ;  the  ordinary,  that  of  the  comple- 
mentary system  corresponding  to  the  same  angle.     The  colours  of  the  two  images  are  thus  seen  in  circumstances 
the  most  favourable,  being  finely  contrasted  and  brought  side  by  side,  so  as  to  be  capable  of  the  nicest  comparison. 
It  is  when  the  vernier  D  reads  45°,  or   the  plane  of  incidence  is  45°,  inclined  to  that  of  primitive  polarization, 
that  the  contrast  of  the  two  images  is  at  its  maximum,  the  tints  in  the  extraordinary  image   being  then  most 
vivid,  and  those  in  the  ordinary'  free   from  any  mixture  of  white  light.     In  general,  if  A  represent  the  light  of 
the  extraordinary  image  in  the  position  above  mentioned,  and  a  the  angle  read  off  on  the  vernier  B,  in  any  other 
position  of  the  plane  of  incidence,  the  two  images  in  this  new  position  (for  the  same  angle  of  incidence)  will  be 
represented  respectively  by 

A  .  (sin  2  a)-,  and  1  —  A  (sin  2  a)- 

that  is,  by  A  .  (sin  2  a)2,  and  (cos  2  «)-  -f  (1  -  A)  .  (sin  2  a)'. 

The  former  of  these  expressions  indicates  a  ray  whose  tint  is  represented  by  A,  and  its  intensity  by  (sin  2  a)2  ;  the 
latter,  a  complementary  tint  1  —  A  of  the  same  intensity,  diluted  with  a  quantity  of  white  light,  whose  intensity 
is  represented  by  (cos  2  a)*. 

932.          These  expressions  represent  with  great  fidelity  the  tints  of  both  images,  the  intensity  of  the  extraordinary,  and 
Agreement    the  apparent  degree  of  dilution  of  the  ordinary  one  ;  and  since  a  ray  A  polarized  in  a  plane  making  an  angle  2  a 
tha  for"    with  the  principal  section  of  the  doubly  refracting  prism,  would  be  divided  between  the  extraordinary  and  ordinary 
M"  Biot's     'ma£Te  m  the  ratio  of  (sin  2  a)1 :  (cos  2  a)8,  it  follows,  that  if  we  regard  the  pencil  at  its  emergence  from  the  cry- 
hypothesis,   stallized  plate  as  composed  of  two  portions,  one  (=  A)  polarized  in  the  above  named  plane,  the  other  (=  1  —  A) 
preserving  its  primitive  polarization,  the  two  pencils  formed  by  the  doubly  refracting  prism  will   be  composed 
as  follows: 

Extraordinary  image.  Ordinary  image. 

1st.  From  the  pencil  A A  (sin  2  «)"  A  .  (cos  2  a)* 

2d.   From  the  pencil  (1  -  A) 0  1  -  A 


Sum    A  (sin  2  «)*         1  -  A  +  A  .  cos  2  a' 

=  1  -  A  .  (sin  2  a)> 

Office  of  the  which  are  identical  with  those  above.     Thus  we  see,  that  the  facts  are  so  far  perfectly  conformable  to  M.  Biot's 
doubly         hypothesis  of  movable  polarization,  and   that  we  are  even   necessitated  to   admit  it,  provided   we  take  it  for 
prisrn'or"     granted,  that  the  rings  exist  actually  formed  and  superposed  in  the  pencil  emergent  from  the  crystallized  lamina, 
tourmaline.  a»d  that  the  office  of  the  doubly  refracting  prism  is  merely  to  analyze  the  emergent  pencil,  and  separate  the  two 
sets  from  each  other.     But  if  the  objection  mentioned  above  against  that  doctrine  be  really  well  founded,  this 
assumption  cannot  be  correct,  and  we  are  then  driven  to  conclude,  that  the   doubly  refracting  prism,  or  tourma- 
line, or  glass  reflector,  interposed  between  the  eye  and  the  crystallized  plate,  performs  a  more  important  office 
than  merely  to  separate  the  tints  already  formed;  and  that,  in  fact,  they  are  actually  produced  by  its  action, — the 
crystallize  I  plate  only  preparing  the  rays  for  the  process  they  are  here  finally  to  undergo. 

933.  To  explain  how  this  may  be  conceived  to  happen  will  form  the  object  of  another  Section.  Meanwhile  we  will 
here  only  add,  that  the  transition  from  uniaxal  to  biaxal  crystals  is  readily  made.  We  have  only  to  consider,  that 
by  varying  the  angle  of  incidence,  (the  line  bisecting  the  angle  between  the  optic  axes  being  supposed  perpen- 
dicular to  the  plane  of  the  plate,)  we  cross  the  rings  in  a  line  passing  through  their  centre  of  symmetry  O,  fig.  183, 
and  makincr  an  ansrle  with  their  princioal  diameter  PP7.  eaual  to  the  angle  read  off  on  the  vernier  B,  and  that 
ny  turning  the  plate  in  its  own  plane,  or  varying  the  angle  read  off  by  the  vernier  c,  we  in  effect  make  the  system 
traversed  pass  through  the  successive  states  represented  in  fig.  179,  180,  181,  182,  changing,  not  the  tint,  but 
the  intensity  of  the  extraordinary  image. 


LIGHT.  527 

Light.          When  the  doubly  refracting  prism  is  turned  in  its  cell,  the  tints  grow  more   dilute,  and  when  placed  in  an    part  jy 
•ps/ofc^  azimuth  a,  that  is,  when  its  principal  section  is  placed  in  the  plane  of  incidence,  both  images  are  colourless,  but  i_j—      _< 
of  unequal  brightness.     This  accords  with  M.  Biot's  doctrine  of  movable  polarization  ;  for  if  we  grant  that  the       934. 
pencil  A  is  polarized  in  a  plane  making  an  angle  2  a  with  that  of  primitive  polarization,  it  will  make,  now,  an  Effect  of 
angle  =  a  with  that  of  the  principal  section  of  the  prism,  and  A  .  (sin  a)a  will  be  that  part  of  the  extraordinary  turning  the 
image  arising  from  the  pencil  A  ;  on  the  other   hand,   the  pencil   1  —  A  retaining  its  original   polarization,  P1 
(1  —  A)  .  sin  a8  will  be  the  portion  of  the  extraordinary  image  produced  by  it  in  the  new  position  of  the  prism, 
and  the  sum,  or  the  whole  image,  will  be  simply  1   x  sin  ae,  which  being  independent  of  A,  or  of  the  tint, 
indicates  that  the  image  is  colourless.     In  the  same  manner  it  may  be  shown,  that  the  ordinary  image  will  equal 
1  X  cos  a1,  and  their  intensities  will,  therefore,  be  to  each  other  as  sin  a*  to  cos  a8,  and  will  be  equal  at  45°  of 
azimuth  ;  all  which  is  conformable  to  fact. 

The  motion  of  the  prism  in  its  cell  corresponds  to  a  rotation  of  the  posterior  tourmaline  in  its  own  plane  in       935. 
the  tourmaline  apparatus.     The  general  appearance  presented  by  the  rings  of  a  single  axis,  when  this  rotation  is  Effect  of 
not  a  precise  quadrant,  is  represented  in  fig.  191,  and  the  succession  of  changes  being  as  follows  :  At  the  first  tu 
commencement  of  the  rotation  the  arms  of  the  black  cross  appear  to  dilate  ;   they  grow  at  the  same  time  fainter,  about  on 
and  segments  of  the  complementary  rings  appear  in  them,  whose  bright  intervals  correspond  to  the  dark  ones  of  each  other. 
the  primary  set,  their  red  to  the  green  portions  of  that  set,  and  vice  versa.     The  junction  of  the  two  sets  is  marked  Fig.  191. 
by  a  faint  white  or  undecided  tint.  As  the  rotation  proceeds,  the  primary  segments  contract  in  extent,  and  become 
more  diluted  with  white,  while  the  secondary  extend,  and  grovv  more  decided  ;  at  the  same  time  the  centre  of  the 
system  grows  gradually  bright,  and  when  the  rotation  has  attained  90°,  the  whole  has  assumed  the  appearance 
in  fig.  188.     The  phenomena  are  precisely  analogous  in  the  rings  of  biaxal  crystals.     The  least  deviation  from 
exact  rectangularity  in  the  tourmalines   gives  rise  to  complementary  segments  in  the   dark  hyperbolic  curves 
answering  to  the   arms  of  the  black  cross,  and  to  a  corresponding  dilution  and  contraction   of  the  primary 
segments,  which  at  last  disappear  altogether  in  the  undistinguishable  whiteness  of  a  pair  of  white  hyperbolas 
precisely  similar  to  the  black  ones  of  the  primary  rings  in  their  perfect  state. 

Hitherto  we  have  considered  the  rings  as  so  narrowed  by  the  thickness  of  the  plate,  as  to  be  all  contracted       936. 
within  a  compass  round  the  poles  which  the  eye  can  take  in  at  once ;  but  if  the  thickness  be  greatly  diminished,  Tints  Pr°- 
this  will  no  longer  be  the  case  ;  and,  instead  of  rings  of  a  distinguishable  form,  we  shall  see  only  broad  bands  ^ce'|h|)^ 
of  colour  extending  to  great  distances  from  the  poles,  and  even  visible  when  the  axes  themselves  are  so  much  f\^es  at 
inclined  to  the  surfaces  of  the  plate  as  to  be  quite  out  of  sight ;  or  even  when  the  axes  actually  lie  in  the  plane  great  dis- 
of  the  plate.     This  is  the  case  with  the  lamina;  into  which  sulphate  of  lime  readily  splits ;  the  axes  lie  in  their  lances  fronv 
plane,  so  that  to  see  the  rings  in  them,  we  must  form  artificial  surfaces  perpendicular  to  the  lamina,  a  difficult  tne  a*es- 
and  troublesome  operation,  from  the  extreme  softness  and  fissile  nature  of  the  substance.     The  phenomena  of 
the  colours  of  this  crystal  were  early  studied,  and  almost  of   necessity  misconceived,   till   Dr.  Brewster,  by 
exhibiting  the  real  axes,  showed  that  they  form  only  a  particular  case  of  the  general  phenomenon  we  have  already 
dwelt  on. 

Adhering  to  the  denominations  employed  in  Art.  885 — 888,  let  us  call  the  plane  containing  the  two  axes,  the      937. 
section  A  ;    that  perpendicular  to  it,  and  passing  through  the  line  which  bisects  the;r  lesser  included  angle,  the  Phenomena 
section  B  ;  and  that  which  similarly  passes  through  the  line  bisecting  their  greater  included  angle,  and  is  perpen-  "^  '^e 
dicular  to  both  the  others,  the  section  C.     If  the  crystal  have  but  one  axis,  the  sections  A  and  B  pass  through  it, 
and  C  is  at  right  angles  to  it.     Then  if  the  lamina  contains  both  axes,  its  plane  will  be  that  of  the  section  A,  and 
the  other  two  sections  will  intersect  it  in  two  lines  (B  and  C)   at  right  angles  to  each  other.     Conceive,  now,  a 
polarized  ray  to  pass  through  such  a  lamina  at  a  perpendicular  incidence.     Then  if  the  plane  of  polarization 
coincide  with  either  of  the  sections  B  and  C,  its  polarization  will  be  undisturbed,  and  the  whole  of  the  trans- 
mitted light  will  pass  into  the  ordinary  image.     But  if  the  plate  be  turned  round  in  its  own  plane,  the  extra- 
ordinary image  will  reappear  and  become  a  maximum  at  every  45°  of  the  plate's  rotation ;    and  if  it  be  suffi- 
ciently thin,  will  exhibit  some  one  of  the  colours  of  the  rings,  and  the  tints  will  descend  regularly  in  the  scale  as 
the  thickness  is  increased,  the  thickness  being  a  measure  of  the  tint,  conformably  to  the  general  law  in  Art.  907, 
of  which  this  is  only  a  particular  case. 

When  two  such  plates  are  laid  together,  with  their  sections  B  and  C  corresponding,  it  is  evident  that  they  are       938. 
in  the  same  relation  as  if  they  formed  part  of  one  and  the  same  crystal ;   and  we  might  therefore  expect  to  find  Phenomena 
what  really  happens,  viz.  that  such  a  compound  plate  polarizes  the  s'ame  tint  that  a  single  plate  equal  to  the  sum  °j 
of  the  thicknesses  would  do.     But  if  they  be  crossed,  i.  e.  laid  so  together  that  the  section  B  of  the  one  shall  Prpeesnad';cau_ 
coincide  with  the  section  C  of  the  other,  M.  Biot  has  shown  that  the  tint  polarized  is  that  due  to  the  difference  iar  mci- 
of  their  thicknesses.     If,  therefore,  this  difference  be  exactly  nothing,  the  crossed  plates  will  be  exactly  neutra-  dence. 
lized,  at  least  at  a  perpendicular  incidence,  and  that  whatever  be  their  thickness.     (To  procure  two  plates  of 
exactly  the  same  thickness,  we  have  only  to  choose  a  clear  and  truly  parallel  plate  terminated  by  fresh  surfaces  of 
fissure,  and  break  it  across.) 

When,  however,  the  incidence  is  not  perpendicular,  such  a  compound  plate  as  described  will  still  exhibit  colours       939. 
which  vary  in,  apparently,  a  very  irregular  manner  as  the  incidence  changes,  and  with  different  degrees  of  Phenomena 
rapidity  in  different  planes.     The  tourmaline  apparatus  here  renders  signal  service  in  rendering  the  law  of  these 
tints,  at  first  sight  extremely  puzzling,  a  matter  of  inspection.     When  such  a  crossed  plate  is  placed  between  the 
tourmalines,  crossed  at  right  angles,  it  exhibits  the  singularly  beautiful  and  striking  phenomenon  represented  in 
fig.  192,  in  which  the  tints  are  those  of  the  reflected  scale  of  Newton,  the  origin  being  in  the  black  cross.     If  the  Y\<,  192 
tourmalines  be  parallel,  the  complementary  colours  are  produced  with  equal  regularity,  as  in  fig.  193.     If  the  f;»  193 
compound  crystal  be  turned  round  in  its  own  plane,  the  figures  turn  with  it,  but  undergo  no  change  other  than 
an  alternation  of  intensity,  being  at  a  maximum  of  brightness  when  the  arms  of  the  cross  are  parallel  and 


528  LIGHT. 

v  ^g1"^      perpendicular  to  the  plane  of  original  polarization,  and  vanishing  altogether  when  they  make  angles  of  45°  with     Tart  IV. 
**  that  plane.     If  the  plates  be  not  crossed  exactly  at  right  angles,  or  he  not  precisely  of  equal  thickness,  other  v— V*"' 
phenomena  arise  which  it  is  easier  for  the  reader  to  produce  for  himself  than  to  read  a  detailed  account  of.     The 
same  may  be  said  of  the  very  splendid  but  complicated  phenomena  produced  by  crossing  two  equally  thick 
plates  of  biaxal  crystals,  such  as  mica,  topaz,  &c.  having  the  section  A  at  right  angles  to  their  surfaces. 
940.  Regarding,  however,  at  present  only  the  tint  produced  at  a  perpendicular  incidence,  it  is  found  that  when  any 

number  of  plates  of  one  and  the  same  medium,  of  any  thicknesses,  are  superposed  with  their  homologous  sections 
'.'""the  su"-  corresponding,  the  tint  polarized  is  that  due  to  the  sum  of  their  thicknesses  ;  but  when  any  one  or  more  of  them 
perposition  nave  their  sections  B  and  C  at  right  angles  to  the  homologous  sections  of  the  others,  the  tint  is  that  due  to  the 
of  similar  sum  of  the  thicknesses  of  those  placed  one  way,  minus  the  sum  of  those  of  the  plates  placed  the  other 
plates.  way.  In  algebraical  language,  if  we  call  t,  t',  t'1,  &c.  the  thicknesses,  and  regard  as  negative  those  of  the  plates 
laid  crosswise,  the  tint  T  polarized  by  the  system  will  be  that  due  to  the  thickness  <-)-<'  -|-  <"-(-  &c. 

When  the  ray  is  made  to  traverse  a  plate  of  quartz,  zircon,  carbonate  of  lime,  or  any  other  uniaxal  crystal  cut  so 
Drod  °f  d"U  aS  <0  conta'n  l^e  ax's  °f  double  refraction,  the  same  law  of  the  tints  holds  good,  the  tint  T  being  proportional  to 
the  thickness  t  of  the  plate,  and  for  any  given  plate  we  have  T  =  k  t,  k  being  a  constant  depending  on  the  nature 
milar  plates  °f  tne  plate.     Now,  if  several  plates  of  different  uniaxal  crystals  be  superposed,  of  which  t,  t',  &c.  are  the  thick- 
nesses, and  if  a  negative  value  of  t  be  supposed  to  denote  a  transverse  position  of  the  axis  of  the  plate,  the 
resultant  tint  will  be  represented  by 

T  =k  t  +  k'tl  +  k"  if'  +  &c. 

942.  In  this  equation,  if  the  plates  be  all  of  one  substance,  k,  k',  &c.  are  all  alike ;  but  if  they  be  different,  k  is 
Opposite      to  be  regarded  as  a  negative  quantity  for  all  such  crystals  as  belong  to  M.  Biot's  repulsive  class,  (Art.  803,)  such 

1  °[      as  carbonate  of  lime  ;  and  positive  for  all  such  (quartz,  for  instance)  which  belong  to  his  attractive  class.  Thus, 
positive  and  eac''  term  in  the  above  equation  may  change  its  sign  from  two  causes,  either  from  a  change  in  the  nature  of  the 
negative       crystal,  or  from  a  change  of  90°  in  its  azimuth. 
crystals.  The  above  is  only  a  particular  case  of  a  more  general  law  which  maybe  thus  announced, — The  tint  ultimately 

943.  produced  is  proportional  to  the  interval  of  acceleration  or  retardation  of  the  ordinary  ray  on  the  extraordinary, 
after  traversing  the  whole  system ;  the  partial  acceleration  or  retardation  in  each  plate  being  proportional  to  the 
length  of  thepath  described  within  the  plate,  multiplied  by  the  square-of  the  sine  of  the  angle  which  the  transmitted 
ray  makes,  internally,  with  the  optic  axis  of  the  plate,  if  it  have  but  one  axis,  or  to  the  product  of  the  sines  of  its 
inclination  to  either,  if  it  have  two ;   and  this  law  holds  good  for  all  positions  of  the  plates,  and  all  arrange- 
ments of  them  one  among  the  other.     Thus  (to  instance  its  application)  in  the  case  of  two  similar  and  equal 
plates  crossed  at  right  angles ;  by  the  laws  of  polarization,  the  ray  which,  after  its  transmission  through  the  first 
plate  is  ordinary,  is  refracted  extraordinarily  by  the  second,  and  vice  versd ;  thus  the  two  rays,  on  entering  the  second 
plate  exchange  velocities ;  and,  therefore,  when  finally  emergent,  since  the  thickness  of  the  second  is  equal  to 
that  of  the  first,  the  one  ray  will  have  lost  ground  on  the  other  in  its  second  transmission  just  as  much  as  it 
gained  it  in  its  first;  and  thus  the  interval  of  retardation  and  the  tint  will  be  reduced  to  nothing. 

944.  From  this  it  appears,  that  if  two  uniaxal  plates  cut  at  right  angles  to  the  axis  be  superposed,  and  adjusted 
Supcrposi-    so  as  to  have  their  axes  precisely  coincident,  the  system  of  rings  will  have  their  diameters  diminished  if  the 

plates  be  both  attractive  or  both  repulsive  ;   but  enlarged,  if  their  characters  be  opposite.     The  experiment  is 
?i'»lit!a^i<'les  ratner  delicate  ;    but  if  made  with  care,  placing  the  plates  on  one  another  with  soft  wax,  and  adjusting  their 
to°theit         surfaces  by  pressure  to  the  exact  position,  it  succeeded  perfectly  in  the  hands  of  Dr.  Brewster. 
axes.  This  affords  a  means,  independent  of  any  measurement  of  the  separation  of  the  ordinary  and  extraordinary 

945.  pencils,  of  ascertaining  whether  an  uniaxal  crystal  be  attractive  or  repulsive ;   for  if  its  rings  be  dilated  by 
Method  of    combining  it  with  a  thin  plate  of  carbonate  of  lime,  cut  at  right  angles  to  the  axis,  it  is  positive  ;   if  contracted, 
whethe'r'a'1''  neSat've-     A  simpler  and  readier  method  still  is  to  fasten  on  a  plate  of  the  substance  under  examination,  so  cut 
crystal  be     as  to  show  the  rings,  a  plate  of  sulphate  of  lime  of  moderate  thickness,  and  then,  interposing  it  between   the 
positive  or    tourmalines,  to  turn  it  about  in  its  own  plane.     A  position  will  be  found  where  the  rings  are  unaltered.     In  this 
negative.      situation  the  section   B  or  C  of  the  sulphate  of  lime  is  in  the  plane  of  primitive  polarization.     If  the  com- 
pound plate  be  turned  45°  from  this  situation,  it  will  now  be  observed  (if  the  thicknesses  of  the  two  plates  be 
properly  proportioned)  that  the  rings  in  two  opposite  quadrants  are  entirely  obliterated ;   and  that  in  the  other 
two  they  are  removed  to  a  much  greater  distance  from  the  centre,  forming  segments  of  larger  circles,  much  closer 
together  ;    and  in  which  the  tints,  instead  of  commencing  from  the  centre,  commence  from  a  black  interval 
between  two  adjacent  white  rings  in  the  midst  of  the  system,  and  thence  descend  in  the  scale  both  inwards  and 
outwards.      In  this  state  of   things,  the  position  of  the  sulphate  of   lime,  with   respect  to  the  tourmalines, 
must  be  carefully  noted;  and  the  crystallized  plate  being  detached,  a  plate  of  carbonate  of  lime,  (perpendicular 
to  its  axis,)  or  of  any  other  known  uniaxal  crystal,  must  be  substituted  for  it ;    and  the    sulphate    of   lime 
replaced  in  the  same  position.     If,  then,  it  be  found,  that  the  same  two  quadrants  of  the  rings  are  obliterated  in 
this,  as  in  the  former  case,  and  the  new  set  of  rings  in  the  other  quadrants  be  also  similarly  situated,— then 
the  crystal  examined  is  of  the  same  character  as  the  carbonate  of  lime,  or  other  crystal  used  as  a  standard  of 
comparison ;  but  if,  on  the  other  hand,  the  quadrants  where  the  rings  were  obliterated  in  the  former  case  be 
those  where  the  new  rings  are  formed  in  the  latter,  then  the  characters  of  the  two  substances  are  opposite.     If 
the  crystallized  plate  be  too  thin,  or  of  too  feeble  polarizing  power  to  exhibit  these  phenomena  with  necessary 
distinctness,  we  must  place  it  in  azimuth  45°  on  the  divided  apparatus  described  in  a  former  article  (929 ;)  and, 
fixing  conveniently  in  the  polarized  beam  a  very  thin  plate  of  sulphate  of  lime  also  in  azimuth  45°,  ascertain, 
by  making  the  crystal  revolve,  whether  its  tints  have  been  raised  or  depressed  in  this  plane  by  the  action  of  the 
sulphate ;  then,  removing  the  crystal,  replace  it  with  a  standard  one,  and  repeat  the  observation  without  touching1 


LIGHT.  529 

Light.       the  sulphate.     If  both  crystals  have  their  tints  raised,  or  both  depressed,  their  characters  are  similar  ;  it  they  be     Part  IV. 
— v"™*'  contrarily  affected,  dissimilar.     An  analogous  mode  of  observation  applies  to  biaxal  crystals.  \~+v~*~s 

§  VIII.   On  the  Interferences  of  Polarized  Rays. 

In  repeating  the  experiments  of  Dr.  Young  on  the  law  of  interference  it  occurred  to  M.  Arago,  that  it  .946. 
would  be  worth  while  to  examine  whether  the  state  of  polarization  of  the  interfering  rays  would  cause  any  Origin  °f 
modification  in  the  phenomena.  The  experiment  was  easy  in  the  case  where  both  rays  had  the  same  polarization, 
being,  in  fact,  the  ordinary  case  ;  but  when  the  interfering  rays  were  required  to  have  a  different  state  of  pola- 
riz.ition,  it  will  easily  be  conceived  that  it  must  be  a  matter  of  great  delicacy  and  difficulty  to  superadd  this 
condition  to  the  others  called  for  by  the  nature  of  the  case,  which  requires  that  the  interfering  rays  should 
emanate  at  the  same  instant  from  a  common  origin,  and  should  have  executed  the  same  precise  number  of 
undulations  or  periods  (within  a  very  few  units)  between  their  origin  and  the  point  where  their  interference  is 
observed.  For  it  is  not  possible  to  change  the  state  of  polarization  of  a  ray  without  either  altering  its  course, 
or  transmitting  it  through  some  medium  in  which  more  or  fewer  undulations  are  executed  in  the  same  space. 
The  joint  ingenuity  of  himself  and  M.  Fresnel,  who  was  associated  with  him  in  this  interesting  inquiry,  how- 
ever, soon  found  means  of  obviating  the  difficulties  and  delicacies  of  the  subject,  and  the  results  of  their  expe- 
riments have  been  embodied  by  them  in  the  following  laws  : 

1.  That  two  rays  polarized  in  one  and  the  same  plane  act  on  or  interfere  with  each  other  just  ax  natural      947. 
rays,  so  that  the  phenomena  of  interference  in  the  two  species  of  light  are  absolutely  the  same.  Laws  of '"" 

2.  That  two  rays  polarized  in  opposite  planes  (i.  e.   at  right  angles  to    each   other)  have  no   appreciable  ^  ^icAi  d 
action  on  each  other,  in  the  very  same  circumstances  where  rays  of  natural  light  would  interfere  so   as   to  light. 
destroy  each  other.  948. 

3.  That  two  rays  primitively  polarized  in  opposite  planes  may  be  afterwards  reduced  to  the  same  plane  ofpola-       949. 
ritation,  without  acquiring  thereby  the  power  of  interfering  with  each  other. 

4.  That  two  rays  polarized  in  opposite  planes,  and  then  reduced  to  similar  states  of  polarization,  interfere       950. 
like  natural  rays,  provided  they  belong  to  a  pencil  the  whole  of  which  was  primitively  polarized  in  one  and  the 

tame  plane. 

5.  In  the  phenomena  of  interference  produced  by  rays  which  have  undergone  double  refraction,  the  place  of  the      951. 
coloured  fringes  is  not  alone,  determined  by  the  difference  of  routes  or  velocities,  but  that  in  certain  circumstances 

a  difference  of  half  an  undulation  must  be  allowed  for. 

Such  are  the  laws  of  interference  of  polarized  pencils,  as  stated  by  Messrs.  Arago  and  Fresnel.     We  use  in       952, 
their  enunciation,  and  indeed  throughout  the  sequel  of  this  part  of  the  doctrine  of  Light,  the  language  of  the 
undulatory  system,  as  really  the  most  natural,  and  adapting  itself  with  the  least  violence  and  obscurity  to  the 
facts.     The  reader  may,  if  he  please,  substitute  that  of  the  corpuscular  hypothesis  and  the  Newtonian  fits,  super- 
adding  that  of  a  rotation  of  the  luminous  molecules  about  their  axes,  with  M.  Biot ;   or  simply  content  himself 
with  a  bare  enunciation  of  facts,  and  with  general  terms  expressive  of  the  existing  conditions  of  periodicity, 
without   much  trouble,  and  only  a  little  circumlocution,  but  with  a  great  sacrifice  of  clearness  of  conception. 
With  respect  to  the  laws  themselves,  the  first  is  easily  verified  ;    we  have  only  to  repeat  any  of  the  experiments  Experimen- 
on  the  interference  of  rays  emanating  from  a  common  origin,  described  in  our  section  on  that  subject,  substi-  tal  verifica- 
tuting  polarized  instead  of  natural   light,  and  the  results  will  be  precisely  similar,  and  that  in  whatever  plane  ''<>"  of  the 
the   light   be    polarized.      Rays,  then,  polarized  in  the  same  plane,   interfere  as  natural   rays  under  similar  first  '**'• 
circumstances. 

The  verification  of  the  second  law  is  more  difficult  and  delicate.     The  conditions  of  the  production  of  colours       953. 
by  interference  require  that  the  interfering  rays  should  emanate  simultaneously  from  a  common  origin,  or  form  Difficulties 
parts  of  one  and  the  same  wave  proceeding  therefrom  as  a  centre  ;  and  should  have  performed,  at  the  point  P*ct 
where  their  interference  is  examined,  the  same  number  of  undulations  in  their  respective  routes,  within   a  very 
few  units.     Now  at  their  leaving  their  origin  they  conld  not  be  otherwise  than  in  the  same  state  of  polarization  ; 
and  as  they  are  required  to  arrive  at  the  point  of  interference  in  opposite  states,  a  change  of  polarization  must 
be  operated  on  one  or  both  rays,  either  by  reflexion,  transmission,  or  double  refraction,  after  leaving  their  origin, 
and   that  without  altering,  more  than  by  a  few  undulations,  the  difference   of   their  routes.     Now,  when  we 
consider  how  minute  a  quantity  an  undulation  is,  it  is  easy  to  conceive  the  delicacv  required  in  adjusting  the 
parts  of  any  apparatus  constructed  for  this  purpose,  or  the  peculiar  contrivances  which  must  be  resorted  to  to 
render  such  extreme  ami  almost  impracticable  nicety  unnecessary. 

Several   ingenious  and  elegant  methods  of  making  the  experiment  have  been  devised  by  the  authors   last       954. 
named,  of  which  we  shall  content  ourselves  with  stating  one  or  two  of  the  easiest  and  most  satisfactory.     And,  Verifica- 
first,   the  origin  of  the  interfering  rays  being  the  image  of  the  sun  at  the  focus  of  a  small  lens,  as  we  shall  ''on  of  'I* 
suppose  it  throughout  this  section,  (unless  the  contrary  be  expressly  said,)  it  is  evident  that  if  we  interpose  secon(1  law- 
between  the  eye  and  this  image  a  rhomboid  of  Iceland  spar,  there  will  be  formed  two  images  separated  from 
each  other   by  a  space  which   will   be  greater  the  thicker  is  the   rhomboid  ;   but  which  will  always   (unless 
extremely  thick  rhomboids  be  used)  be  very  small ;  so  that  the  single  luminous  point  will  now  be  resolved  into 
two,  very  near  each  other,   and  which,  by  the  laws  of  polarization,  send  to  the  eye  rays  polarized  in  opposite 
ptanes.     But  in  this  disposition  of  things,  the  condition  of  near  equality  of  routes  is  subverted ;  for  the  ordinary 
and  extraordinary  pencils  pursue  different  paths  within   the  crystal,  and  with  very  different  velocities ;  so  that 
a  difference  will  thus  arise  in  the  total  number  of  undulations  executed  by  each,  sufficient  to  destroy  all  evidence 

VOL.  iv.  3  2 


530  LIGHT 

Light,     of  interference  by  the  production  of  coloured  fringes.     To  obvial  >  this  diCiculty,  M.  Fresnel  sawed  in  half  a 

V"~"V~^/  rhomboid  of  Iceland  spar,  the  two  halves  of  which  must  of  necessity  have,  at  their  line  of  separation  and  its  '^ 

M-Fresne^'ts  immediate  confines,  precisely  equal  thicknesses.     These  halves  he  placed  one  on  the  other,  only  turning  one 

wi?h"'    "   9®°  round   >n  azimuth,  so  as  to  have  their  principal  sections  at  right  angles.     In  this  state,  a  pencil  entering 

bisected       them  nearly  at  the  intersection  of  the  planes  of  separation  would  at  its  final  emergence  be  divided,  not  into  four, 

rhomboid,    but  into  two  only,  (see  Art.  879,)  the  ray  ordinarily  refracted  in  the  first  half  having  undergone  extraordinary 

refraction  in  the  second,  and  vice  versd.     The  two  rays,  therefore,  have  exchanged  velocities  and  directions,  in 

the   second  transmission  ;    and,  therefore,  when  emergent,  will  have  described  exactly  equal  paths  with  equal 

velocities  in  each  respectively,  and  will  differ  only  in  their  states  of  polarization,  which  will  be  at  right  angles 

to  each  other.     We   have  here,  then,  a  case  in  which  pencils  diverge  from  two  points  side  by  side,  and  in  a 

state  in  all   other  respects  proper  for  interfering ;  nevertheless,  when  we  look  for  the  fringes  which  ought  to 

be  formed  under  such   circumstances,  (and  which  with   natural  light  would  be  seen,  see  Art.  735  and  736,) 

none  are  visible.     Their  absence,  then,  must  be  owing  to  the   opposite  state  of  polarization  of  the  inter- 

rering  rays. 

955.  M.  Arago,  to  make  the  same  experiment,  employed  a  process  independent  of  double  refraction.     Two  fine 
M.  Arago's  sj;ts  were  ma(}e  in  a  thin  plate  of  copper,  through  which  rays  from  the  common  origin  were  transmitted,  and 
wUiTmicd ** f°rmed  fringes  (in  their  natural  state)  when  viewed  by  an  eye  lens  in  the  manner  described,  (Art.  709.)     He 
piles.           now  prepared  two  piles  of  pieces  of  very  thin  mica,  or  films  of  blown  glass  laid  one  on  the  other,  fifteen   iu 

number,  and  then  divided  this  compound  plate  in  half  by  a  sharp  instrument,  so  that  the  halves,  in  the  imme- 
diate neighbourhood  of  the  line  of  division,  could  not  be  otherwise  than  of  almost  exactly  equal  thickness. 
These  piles,  when  exposed  at  an  incidence  of  30°  to  a  ray,  were  found  to  polarize  the  portion  transmitted 
almost  completely.  They  were  then  placed  before  the  slits  so  as  to  receive  and  transmit  the  rays  from  the 
luminous  point  at  precisely  that  incidence,  and  through  spots  which  were  very  near  each  other  in  the  undivided 
state  of  the  pile.  They  were,  moreover,  so  arranged,  (being  set  on  revolving  frames,)  that  the  plane  of 
incidence  could  be  varied  (and  therefore  that  of  polarization)  by  turning  either  round  in  azimuth  without  alter- 
ing its  inclination  to  the  ray,  or  varying  the  spot  through  which  the  ray  passed.  And  it  was  then  found,  that 
when  both  piles  were  placed  so  as  to  polarize  the  rays  in  parallel  planes,  as,  for  instance,  when  both  were 
inclined  directly  downwards,  or  one  directly  down  and  the  other  directly  up — the  fringes  were  formed  as  if  the 
piles  were  away ;  but  where  one  of  the  piles  was  turned  round  the  incident  ray  as  an  axis  through  90°,  and  so 
placed  as  to  polarize  the  rays  transmitted  by  it  at  right  angles  to  the  other,  the  fringes  totally  disappeared,  nor 
could  they  be  restored  by  inclining  either  pile  a  little  more  or  less  to  the  incident  ray  in  the  plane  of  incidence, 
the  effect  of  which  would  be  to  alter  gradually  the  length  of  the  ray's  path  within  the  pile  without  changing 
its  polarization,  and  thus,  to  compensate  any  slight  inequality  which  might  still  subsist  in  (heir  thicknesses. 
In  intermediate  positions  the  fringes  appeared,  but  always  the  more  vividly  the  nearer  the  planes  of  polariza- 
tion approached  to  exact  parallelism,  thus  attaining  their  maximum,  and  undergoing  total  obliteration  at  each 
quadrant  of  the  rotation  of  either  pile,  (the  other  being  at  rest.) 

956.  A  plate  of  tourmaline  carefully  worked  to  exact  parallelism,  and  bisected,  would  answer  equally  well  with  the 
Tourmaline  transparent  piles  to  polarize  the  rays ;  but  the  tourmaline  should  be  selected  of  very  homogeneous  texture,  such 
plates  sub-  are  not  easy  (O   ,neet   with,  though   they  may  be  found  ;    and  in  this  manner  the  experiment  is  perfectly  easy 
*h'tUtUesf0r  an(*  satisfactory-     One  half  the  tourmaline  is  fixed  over  one  aperture,  the  other  movable  in  a  cell  in  its  own 

plane  over  the  other.  The  same  phenomena  will  then  be  observed  by  turning  round  the  movable  tourmaline  as 
with  the  oblique  pile  in  the  last  experiment. 

957.  An  experiment  still  more  simple,  and  equally  conclusive,  is  the  following,  of  M.  Fresnel.     He  placed  before 
M.Fresuel's  the  sheet  of  copper  (having,  as  before,  two  narrow  slits  in  it  very  near  each  other)  a  single  thin  parallel  lamina 
fundamen-    of  suipnate  of  lime.     Now,  as  this  body  possesses  double  refraction,  each  pencil  would  be  divided  into  two — 
ment           an  ordinary  and  an  extraordinary  one — which,  according  as  they  emanate  from  the  right  or  left  hand  slit,  we 
Analysis  of  will  term  R  o,  R  e,  and  L  o,  L  e.     If  natural  light  be  used  to  illuminate  the  slits,  these  pencils  will  be  of  equal 
the  pola-     intensity,  but  those  marked  e  will  be  polarized  oppositely  from  those  marked  o.     We  may  then  form  four  *om- 
rizedtint*.    binations  :   1.  Ro  may  interfere  with  L  o  ;  2.   R  e  may  interfere  with  Le,-    3.  R  o  with  Le;  4.  R  e  with  L  a. 

Now  of  these,  Roand  L  o  are  similarly  polarized,  and  they  have  described  equal  paths  with  equal  velocities; 
therefore,  supposing  them  capable  of  interference,  they  will  give  rise  to  a  set  of  fringes  corresponding  exactly 
to  the  middle  of  the  line  joining  the  two  slits,  or,  as  we  may  express  it,  in  the  axis  of  the  apparatus.  The 
same  may  be  said  of  R  e  and  L  e.  These  two  sets  of  fringes  will  therefore  be  superposed,  and  appear  as  one  of 
double  intensity.  Again,  R  o  may  be  combined  with  L  e  ;  but  as  these  two  rays  have  traversed  the  sulphate  in 
different  directions  and  with  different  velocities,  those  rays  of  each  pencil  which  meet  in  the  axis  will  differ  by 
too  many  undulations  to  produce  colour ;  and  if  the  pencils  interfere,  the  place  of  the  fringes  will,  instead  of 
the  axis,  be  shifted  towards  the  side  where  the  pencil  has  the  greatest  velocity,  (Art.  737,)  and  that  the  more, 
the  thicker  the  lamina  of  sulphate,  so  that  if  taken  of  a  proper  thickness,  this  set  of  fringes  may  be  removed 
entirely  out  of  the  reach  of  the  middle  set,  and  should  be  seen  independent  of  it.  In  like  manner,  the  pencil 
R  e  may  interfere  with  L  o,  and  give  rise  to  another  set  of  lateral  fringes ;  but  as  the  ray  which  in  the  former 
combination  was  the  swifter,  in  this  is  the  slower,  this  set  will  lie  on  the  opposite  side  of  the  middle  set,  sup 
posing  it  produced  at  all ;  and  thus  there  should  be  seen  three  sets  of  fringes,  one  bright,  in  the  middle, 
and  two  fainter  on  either  side.  But,  in  fact,  only  one  set  is  seen,  viz.  the  middle  set.  Therefore  the  combina- 
tion of  the  rays  R  o  and  L  e,  L  o  and  R  e,  which  are  polarized  oppositely,  produce  no  fringes,  i.  e.  they  do  not 
interfere. 

But  if  we  cut  the  lamina  in  half,  and  turn  one  half  a  quadrant  round  in  its  own  plane,  these   rays  are  tiitr, 
reduced  to  the  same  polarization ;   and  the  rays  R  o  and  L,o,Re  and  L  e,  which  in  the  former  case  gave  rise  to 


LIGHT. 


531 


Light,      the  central  fringes,  are  now  placed  in  opposite  states  of  polarization  ;  and  it  is  accordingly  found  that  the  central    Part  IV 
•—v*-™"  fringes  have  disappeared  entirely,  and  that  two  lateral  sets  formed  respectively  by  R  o  and  Le,  Re  and  L  o,  ^— -\~~' 
have  started  into  existence.     If  we  turn  the  lamina  slowly  round,  these  will  gradually  fade  away,  and  the  central  Experiment 
reappear  and  become  brighter,  and  so  on  alternately  ;    thus  affording  a  convincing  proof  of  the   truth  of  the  vaned- 
second  of  the  laws  above  enunciated. 

The  experiment  related  by  Messrs.  Arago  and  Fresncl  in  support  of  their  third  law  is  as  follows  :  Resuming       959. 
the  arrangement  of  Art.  955  or  956,  and  placing  the  piles  or  tourmalines  so  as  to  polarize  the  two  pencils  Verification 
oppositely,  let  a  doubly  refracting  crystal  be  placed  between  the  eye  and  the  sheet  of  copper,  with  its  principal  °^  l'le  ""'"' 
.section  45°  inclined  to  either  of  the  planes  of  polarization  of  the  interfering  rays.     Each  pencil  will  then  divide  aw' 
itself  by  double  refraction  into  two  of  equal  intensity,  and  polarized  in  two  planes  at  right  angles,  one  of  which 
is  the  principal  section  itself.     We  ought,  therefore,  to  expect  to  see  two  systems  of  fringes,  one  produced  by 
the  interference  of  the  ordinary  ray  from  the  right  hand  aperture  (R  o)  with  that  of  the  left  (L  o,)  and  the  other 
by  that  of  Re  with  Le;  yet  no  fringes  are  seen.     The  experiment  may  be  varied  by  substituting  for  the  doubly 
refracting  prism  a  tourmaline,  or  pile,  with  its  principal  section  in  azimuth  45°.     This  must  reduce  to  a  common 
polarization  all  the  rays  which  traverse  it,  viz.  the  half  of  each  pencil,  yet  no  fringes  are  seen,  and  therefore  no 
interference  takes  place. 

The  following  experiment  is  adduced  in  the  Memoir  cited  in  support  of  the  fourth  and  fifth  of  the  above      960. 
laws.     A  lamina  of  sulphate  of  lime  is  perpendicularly  exposed  to  a  polarized  pencil  diverging  from  a  minute  ExPeri; 
point,  and  immediately  behind  it  is  placed  a  plate  of  brass  pierced  with  two  very  small  holes  near  together.  ments  ln 
The  principal  section  of  the  lamina  is  to  be  placed  at  an  angle  of  45°  with  the  plane  of  primitive  polarization,  fourth  and 
In  consequence,  from  each  of  the  holes  (right,  R, — and  left,  L)  will  emerge  a  ray  composed  of  two  equal  rays,  fifth  laws. 
R  o  and  R  e,  and  L  o,  L  e  oppositely  polarized,  viz.  at  angles   +  45°  and  —  45°  with  the  plane  of  primitive 
polarization,  which  we  will  suppose  vertical.     In  this  situation  of  things  a  rhomboid  of  Iceland  spar  is  placed 
between   the  two  holes,  and  the  focus  of  the  eye  lens  employed  to  view  the  fringes,  with  its  principal  section 
vertical,  i.  e.  making  again  with  that  of  the  lamina  angles  of  45°  either  way.     Each  of  the  four  rays  then  above 
mentioned  will  be  divided  into  two  equal  rays,  an  ordinary  and  an  extraordinary,  thus  giving  rise  in  all  to  the 
eight  rays 

Roo,  Reo;  Loo,  Leo;  Roe,Ree,-  Loe,  Lee. 

These  rays  are  received  on  the  eye  lens,  and  conveyed  into  the  eye.     Let  us  now  examine  their  respective  route 
and  states  of  polarization. 

First,  then,  the  rays  Ro  and  Re,  after  quitting  the  lamina,  are  parallel;  and  by  reason  of  the  very  small  951. 
thickness  of  it,  may  be  regarded  as  superposed,  being  undistinguishable  from  each  other;  but  they  have 
described  within  the  lamina  different  paths  by  different  velocities,  so  that  on  emerging  they  will  differ  in  phase, 
by  an  interval  of  retardation  proportioned  to  the  thickness  of  the  lamina,  and  which  we  will  call  d,  so  that  a 
being  the  phase  of  the  ray  R  o,  x  -f-  d  will  be  that  of  R  e.  The  very  same  may  be  said  of  L  o  and  L  e.  More- 
over, the  two  rays  of  either  of  these  pairs  respectively  are  oppositely  polarized,  viz.  in  planes  +  45°  and  —  45° 
from  the  vertical.  This  we  may  represent  at  once  thus  :  9 


Ray. 

Ro 
Re 

Lo 
Le 


Phase. 
X 

X  +  d 
X 

x  +  d 


Plane  of  Polarization. 
+  45° 

-  45° 
+  45° 

-  45° 


Next,  the  portions  into  which  either  of  these  rays  is  subdivided,  in  traversing  the  rhomboid,  follow  in  their 
passage  through  it  different  paths,  and  have  different  velocities ;  but  all  which  are  refracted  ordinarily  have  one 
common  direction  and  velocity ;  and  so  of  those  refracted  extraordinarily ;  hence,  between  the  ordinary  and 
extraordinary  rays  here  produced,  will  arise  a  difference  of  phase  which  we  shall  call  £,  so  that  if  x  be  the  phase 
of  any  ordinary  ray,  x  -f-  &  will  be  that  of  the  corresponding  extraordinary  one  ;  and  their  planes  of  polarization 
will  be  opposed,  and  will  form  angles  respectively  =  0  and  90°  with  the  vertical.  Thus  the  circumstances  will 
stand  thus  : 


Ray. 

Roo 
Reo 
Loo 
Leo 


A. 

Phase. 


x 

x  +  d 


Plane  of  Polarization. 
0° 
0° 
0° 
0° 


Ray. 
Roe 
Ree 

L  oe 
Lee 


B. 

Phase. 
X  +  S 

x  -f-  d  -f-  <S 

x  +  c 

x  +  d  +  c 


Plane  of  Polarization. 
90° 
90° 
90° 
90° 


These  eight  pencils  are  all  equal  in  intensity,  and  all  those  contained  in  the  first  set  (marked  A.)  will  meet  in 
one  part  of  the  field  of  view,  while  those  marked  B  (on  account  of  the  thickness  of  the  rhomboid,  which  we 
here  suppose  considerable,  so  as  to  produce  a  sensible,  and  even  a  large  separation  of  the  ordinary  and  extra- 
ordinary pencils)  will  meet  in  another,  distant  from  the  point  of  concourse  of  (A)  by  an  interval  proportional  to 
the  thickness  of  the  rhomboid,  and  which  we  will  here  suppose  so  large  as  to  throw  the  fringes  (if  any)  there 
produced,  entirely  out  of  the  way  of  mixing  with  those  produced  at  the  concourse  of  A.  Let  us  then  consider 
separately,  the  pencils  of  rays  of  the  parcel  A,  and  see  what  interferences  can  take  olace.  And  first,  Roo  may 

3  z  2 


962. 


963. 


532  L  I  G  H  T. 

Light,     combine  with  Loo,  and  since  their  difference  of  phase  is  zero,  they  will  interfere  in  the  axis  of  the  apparatus; 
*—•— y-™^  and  their  planes  of  polarization  being  coincident,  there  is  no   reason  why  fringes  should  not  there  be  pro- 
duced by  their  concourse.     The  same  holds  good  of  the  combination  R  <;  o   and   Leo,  and,  consequently,  there 
will  be  superposed  on  each  other  in  the  axis  two  sets  of  fringes,  producing  one  of  double  brilliancy. 

964.  Next,  R  o  o  may  interfere  with  Leo;    but  there  being  a  constant  difference  of  phases  d  in  favour  of  the  latter, 
the  fringes  produced  by  their  concourse  will  lie  to  the  left  of  the  axis,  by  an  interval  proportional  to  the  thickness 
of  the  lamina  of  sulphate,  and  will  be  seen  separately.     Similarly,  the  concoin  se  of  the  pencils  Reoand  Loo  will 
determine  the  production  of  another  set  of  lateral  fringes ;  but  the  difference  of  phases  d  being  in  this  case  in  favour 
of  the  right  hand  pencil,  this  system  will  be  situated  as  much  to  the  right  of  the  axis  as  the  other  was  to  the  left. 

965.  Thus  in   the   ordinary  image   three  sets  of  fringes  ought  to  be  seen,  and  in  the  extraordinary,  by  a   similar 
reasoning,  as  many.     Now,  in  fact,  this  is  the  case,  and  the  phenomena  are  seen  on  making1  the  experiment  pre- 
cisely as  here  described.     But  it  is  evident  that  the  rays  which  form  the  lateral  fringes,  by  their  interferences,  are 
precisely  those  which,  at  their  leaving  the  sulphate,  had  opposite  polarizations,  but  have  been  afterwards  reduced 
to  similar  polarization  by  the  action  of  the  rhomboid. 

966.  If  instead  of  a  rhomboid  of  sensible  double  refraction  we  substitute  a  plate  of  sulphate  of  lime,  or  of  rock 
Variation      crystal,  so  thin  as  to  produce  no  visible  separation  of  the  pencils,  the  fringes  produced  by  the  pencils  B  will  be 

i     .          superposed  on  those  arising  from  the  interference  of  the  pencils  A,  and  we  should  expect  therefore,  instead  of  six. 

'  '  to  see  three  sets  of  fringes,  the  middle  one  being  still  the  brightest.     But,  in  fact,  we  see  but  one  set,  and  the 

Allowance    lateral  fringes  vanish  altogether.     This  remarkable  result  proves  that  the  colours  resulting  from  the  concourse  of 

of  half  an     the  rays  ordinarily  refracted  by  the  rhomboid,  are  complementary  to  those  resulting  from  that  of  the  extraordinary 

judulation.  ravg  .  an(j  therefore  that  we  must  allow  half  an  undulation  to  be  gained  or  lost  when  we  would  pass  from  one  set 

to  the  other,  precisely  as  in  the  phenomena  of  the  reflected  and  transmitted  colours  of  thin  plates. 

967.  One  of  the  most  important  consequences  of  these  laws,  is  that  they  supply  the  defective  link  in  the  chain  which 
Application  connects  the  doctrine  of  undulations  with  the  colours  of  crystallized  laminae  as  described  in  the  last  section.     It 
ro  the           had  been  already  remarked  (as  we  have  seen)  by  Dr.  Young,  that  the  passage  of  the  ordinary  and  extraordinary  rays 
(•"'•staTlized  with  different  velocities  through  the  crystallized  plate,  would  give  rise  to  that  difference  of  physical  condition  of 
lamina:.'"     tne  ravs  at  tne'r  emergence  which  would  lead  to  the  production  of  colours ;   but  the  difficulty  remained  to  explain, 

not  why  colours  were  produced  in  certain  circumstances,  but  why  they  were  not  produced  in  all,  in  short,  what 
share  the  polarization  of  the  incident,  and  the  analysis  of  the  emergent  rays,  had  in  the  production,  of  the  phe 
nomena. 

968.  To  see  the   nature  of  this  difficulty  more  clearly,  imagine  a   wave  proceeding  from  a  distant  radiant  point 
Why  co-      to  be  incident  on  a  very  thin   crystallized  lamina.     It  will   be  subdivided  into  two,  each  traversing  the  plate 
lours  are       jn  a  different  direction  and  with  its  own  proper  velocity,  and  each  of  them  emerging  parallel  to  its  original  direc- 
tion.    The  incident  wave  will,  therefore,  after  emergence  be  resolved   into  two  parallel  to  each  other,  but  sepa- 

thin  crystal-  r£>ted  by  a  small  interval  equal  to  the  interval  of  retardation.     Now   the  hindmost  of  these  ought,  according 

lized  plate    to  the  law  of  interferences,  to  interfere  with  a  subsequent  wave  of  the   system  to  which  the  foremost  belongs, 

alone.          and  thus  periodical  colours  should  arise  on  merely  looking  against  the  sky  through  such  a  lamina  without  any 

other  apparatus.     Why  then  are  none  seen  ?     To  this  the  law  of  Messrs.  Arago  and  Fresnel  afford  a  satisfactory 

answer.     The  two  systems  of  waves  into  which  the  incident  system   is  resolved  are  oppositely  polarized,  and 

therefore,  though  all  other  conditions  be  satisfied,  incapable  of  interfering. 

969.  To  understand  how  the  colours  of  the  polarized  rings  must  be  conceived  to  be  produced  by  interference,  let  us 
Fig.  194.      take  the  simplest  case  when  a  polarized  ray,  A  B,  fig.  194,   is  incident  on  any  thin   crystallized  plate  B,  whose 
Explanation  principal  section  is  45°  inclined  to  the  plane  of  primitive  polarization.     Let  A  be  the  system  of  waves  which 
Smire'of'tne  constitutes  the  incident  ray ;   then  in  its  passage  through  the  crystallized  lamina  it  will  be  divided  into   systems 
polarized      °  antl  E  of  equal   intensities,   polarized  in  planes  +  45°  and  —  45°  inclined  to  that  of  primitive  polarization, 
lings.           and  the  one  lagging  a  few  undulations  behind  the  other,  so  as  to  interfere,  as  represented  in  the  fi<mre,  and  con- 
stituting the  parallel  rays  C  F  and  D  G.     Let  these  now   be  received  on,  and   transmitted  through,  a  doubly 
refracting  prism  F  G  H  L  placed  with  its  principal  section  in  the  plane  of  primitive  polarization,  or  45°  inclined  to 
that  of  the  crystallized  lamina.     Then  will  each  of  the  incident  rays  be  again   subdivided,    C  F  into  H  M  and 
I  P,  and  D  G  into  K  N  and  L  Q,  all  of  equal  intensity.     Of  these,  H  M  and  K  N  emerge  parallel,  as  also  K  N 
and  L  Q  respectively.     Now  the  systems  of  waves  O  and  E  which  follow  each  other  at  a  certain  interval  d  will 
continue  to  do  so  in  both  the  refracted  rays,  as  if  they  formed  one  compound  system  ;    so  that  each  of  the  pencils 
H  M  K  N  and  I  P  L  Q  will  consist  of  a  double  system  of  waves  O  e  and  E  e,  O  o  and  E  o  respectively.    The  former 
pair  following  each  other  at  the  interval  d,  and  the  latter  at  the  interval  d  i  J  undulation,  (by  reason  of  the  demon- 
strated fact,  that  in  passing  from  the  ordinary  to  the  extraordinary  system  half  an  undulation  must  be  allowed. 
See  Art.  966.)     Now  as  each   ray  of  these  pairs  respectively  have   similar  polarizations,  viz.  those  of  the  pair 
ordinarily  refracted  (O  o  and  E  o)  in   the  plane  of  the  principal  section  of  the  prism,  and  those  of  the  extra- 
ordinary pair  O  e  and  E  e  in  a  plane  at  right  angles  to  it,  there  is  no  reason  why  interference  should  not  take 
place,  and  the  consequence  must  be,  the  production  of  complementary  colours  in  the  two  pencils  finally  emergent 

corresponding  to  the  intervals  of  retardation  d  and  d  -J ,  which  is  just  what  really  happens. 

m 

970 
Explanation      Conceive  now  another  ray  incident  on  B  in  the  direction  A  B,  but  polarized  in  a  plane  at  right  angles  to  that 

of  the  com-  of  the  ray  considered  in  the  last  paragraph.     Then  this  will  undergo  precisely  the  same  series  of  divisions  an 
jJemenWry  subdivisions  as  the  former.     But  the  intervals  of  retardation  will  be  different;   for  its  plane  of  polarization  when 
*i»ts.  incident  on  B  being  now  related  to  the  plane  of  ordinary  refraction,  as  that  of  the  other  ray  at  its  ,'ncidence  wa» 


LIGHT.  533 

Light.       to  the  extraordinary,  and  vice,  versd,  a  difference  of  half  an  undulation  must  (as  already  explained)  be  admitted     part  jy 
••v  •'• '  in  the  relative  position  of  the  two  systems  of  waves  O,  E,  at  their  emergence,  from  this  cause,  independent  of  v_         _  . 
the  interval  of  retardation  within  the  plate  ;  so  that  if  d  were  the  interval  in  the  former  case,  d  —  %  X  will  be  the 
difference  now,  and,  after  passing'  through  the  prism,  we  shall   have  for  the  intervals  of  retardation  in  the  two 
binary   pencils,    instead  of  d  and  d  +  ^  \  which   they  were   before,  d  —  J  X  and  d.     Hence  the  two  pencils 
will  exchange  colours  when  the  polarization  of  the  incident  light  is  varied  by  a  quadrant,   and  this  is  also 
conformable  to  fact.     If  this  reasoning  be  not  thought  conclusive,  the  reader  is  referred  forwards  to  Art.  983 
and  984. 

Next,  let  the  incident  ray  be  unpolarized.     This  case,  as  we  have  seen  Art.  851,  is  the  same  with  that  of  a       971. 
ray  consisting  of  two  equal  rays  oppositely  polarized,  and  therefore  in  each  pencil  will  coexist,  superposed  on  Why  co- 
each  other,  the  primary  and  complementary  colour  arising  from  either  portion,  which  being  of  equal  intensity  lours  are  not 
will  neutralize  each  other's  colours  and  the  emergent  pencils  will  be  white,  and  each  of  half  the  intensity  of  the  F^uced  by 
incident  beams.     This  then  is  the  reason  (on   this  doctrine)  why  we  see  no  colours  when  the  light  originally  j|°{| 
incident  on  the  crystallized  plate  is  unpolarized 

Thus,  the  theory  of  interferences,  modified  by  tne  principles  above  stated,  affords,  as  we  see,  an  explanation       972. 
of  the  colours  of  crystallized  plates  totally  distinct  from  that  of  movable  polarization.     The  only  delicacy  in   its  M.  Fresnel's 
application  to  all  cases,  lies  in  the  determination  which  of  the  emergent  pencils  must  be  regarded  as  having  its  general rul* 
interval  of  retardation  increased  by  half  an  undulation.     M.  Fresnel  gives  the  following  rule  for  this  essential  Jjj 
point.   (Note  on   M.  Arago's  Report  to   the   Institute  on   a  Memoir  of  M.  Fresnel  relative  to   the  colours  of  to  a]]0^v  fm. 
doubly  refracting  laminae,  Annales  de  Chimie,  vol.  xvii.  p.  80.*)     The  image  whose  tint  corresponds  precisely  to  the  half  un- 
the  difference  of  routes,  is  that  in  which  the  planes  of  polarization  of  its  constituent  pencils  after  having  been  sepa-  dulation 
rated  from  each  other,  are  brought  together  by  a  contrary  motion,  while,  on  the  other  hand,  the  pencils  whose  p'""1  or 
planes  of  polarization  are  brought  to  coincidence  by  a  continuance  of  the  same  motion  by  which  they  were  sepa- 
rated, produce  by  their  reunion  the  complementary  image.     To  understand   this  better,  let  P  C  be  the  plane  of  Fl>=1  '*"*• 
primitive  polarization  projected  on  that  of  the  paper,  to  which  let  us  suppose  the  ray  perpendicular,  C  O  that  of 
the  principal  section  of  the  crystallized  lamina,  and  C  S  that  of  the  principal  section  of  the  doubly  refracting 
prism  ;  then  the  incident  pencil  polarized  in  the  plane  P  P'  will  after  penetrating  the  lamina  be  divided  into  two, 
one  O  polarized  in  the  plane  C  O,  the  other  E  in  the  plane  C  E  perpendicular  to  it.     Now,  C  O  may  always  be 
so  taken  as  to  make  an  angle  not  greater  than  a  right  angle  with  C  P,  and  C  E  so  as  to  have  C  P  between  C  E 
and  C  O  ;  so  that  the  plane  C  P  may  be  conceived  to  open  or  unfold  itself  like  the  covers  of  a  book,  into  C  O  and 
C  E,  one  on  either  side.     Again,  C  S  may  always  be  regarded  as  making  an  angle  not  greater  than  a  right  angle 
with  C  O,  and  when  the  ray  O  resolves  itself  into  two  (O  o  and  O  e)  by,  refraction   at  the  prism,  its  plane  of 
polarization  C  O  may  be  conceived  to  open  out  into  the  two  C  S  and  C  T  at  right  angles  to  each  other,  including 
C  O  between  them ;  and  in  like  manner  the  ray  E  will  resolve  itself  into  two  E  o  and  E  e,  and  its  plane  of  pola- 
rization C  E  will   open  out  into  the  two  C  S  and  C  T,  having  C  E  between  them   in  the  case  of  fig.  195  (a), 
and  into  C  S'  and  C  E  in  that  of  fig.  195  (6)  ;  in  the  former  case  C  T'  is  a  prolongation  of  C  T,  in  the  latter  C  S' 
is  a  prolongation  of  C  S.     The  rays  O  o  and  E  o  then  which   make  up  the  ordinary  pencil,  have,  in  the  case  of 
fig.  £<z),  been  each  brought  to  a  coincident  plane  of  polarization  C  S   by  two  motions  in  contrary  directions,  as 
represented  by  the  arrows,  and  the  extraordinary  ones  O  e  and  E  e  have  been  separated  and  brought  back  to  a 
coincident  plane  by  motions  continued  in  the  same  direction  for  each  respectively.     The  reverse  is  the  case  in 
fig.  b.     In  the  case  then  of  fig.  a  the  colours  of  the  ordinary  pencil   O  o  -f-  E  o  will  be  those  which  correspond 
precisely  to  the  difference  of  routes,  and  those  of  the  extraordinary  one  Oe  +  Ee  will  correspond  to  that  differ- 
ence plus  half  an  undulation,  while  in  that  of  fig.  6  the  reverse  happens.     This  rule  is  empirical,  i.  e.  is  merely  a 
result  of  observation.     It  is  clear  that  the  principle  of  the  conservation  of  the  vis  viva  in  this,  as  in  the  colours 
of  uncrystallized  plates,   requires  that  the  two  images  should  be  complementary  to  each   other,  and  therefore 
half  an  undulation  must  be  gained  or  lost  by  one  or  the  other  pencil,  but  which  of  the  two  is  to  be  so  modified 
we  have  no  means  of  knowing  a  priori. 

This  once  determined,  however,  we  have  no  difficulty  in  deducing  the  formulae  of  intensity  and  other  circum-       0,73 
stances  of  the  phenomena  when  the  azimuth  of  the  crystallized  plate  is  arbitrary,  instead  of  being,  as  we  have 
hitherto  supposed,  limited  to  45°.    The  analytical  expressions  of  the  intensity  of  the  pencils  we  must  reserve  for 
our  next  section. 

§  IX.     Of  the  application  of  the  Undulatory  Doctrine  to  the  explanation  of  the  phenomena  of  Polarized  Light 

and  of  Double  Refraction. 

The  phenomena  of  double  refraction  and  polarization,  as  exhibited  in  the  experiments  of  Huygens  on  Iceland       974^ 
spar,  were  regarded  by  Newton  and  his  followers  as  insuperable  objections  to  the  undulatory  doctrine,  inasmuch  Newton's 
its  it   appeared  to  them  impossible,  by  reason  of  the  quaqudversum  pressure  of  an  elastic  fluid,  to  conceive  an  objections 
undulation  as  having  a  different  relation  to  different  regions  of  space,  or  as   possessing  sides.     "  Are  not,"  says  against  the 
Newton,  "  all  hypotheses  erroneous  in  which  light  is  supposed  to  consist  in  pressure   or  motion  propagated  "1"edc"lat<iry 

*  This  Memoir  was  read  to  tne  Institute,  Oct.  7, 1816.  A  Supplement  was  receired  Jan.  19,  1818.  M.  Arago's  report  on  it  was  read 
June  4, 1821.  And  while  every  optical  philosopher  in  Europe  has  been  impatiently  expecting  its  appearance  for  suven  years,  it  He«  w  yet 
unpublished,  and  is  known  to  us  only  by  meagre  notices  in  a  periodical  Journal. 


534  LIGHT. 

Light,      through  a  fluid  medium?"...  .          "  for  pressures  or  motions  propagated  from  a  shining  body  through  an  uni-     Pan  r' 
v— • "v*~*'  form  medium,  must  be  on  all  sides  alike,  whereas  it  appears  that  the  rays  of  light  have  different  properties  in  v — v~— 

their  different  sides." "To  me,  this  seems  inexplicable,  if  light  be  nothing  else  than  pressure  or  motion 

propagated  through  ether."     Opticks,  book  iii.  quest.  28.     And,  again,  quest.  29  ;   "Are  not  rays  of  light  very 

small  bodies  emitted  from  shining  substances?" "  The  unusual  refraction  of  Iceland  crystal  looks  very 

much  as  if  it  were  performed  by  some  kind  of  attractive  virtue  lodged  in  certain  sides  both  of  the  rays  and  of  the 

particles  of  the  crystal."' "  I  do  not  say  this  virtue  is  magnetical. — It  seems  to  be  of  another  kind.     I  only 

say,  that,  whatever  it  be,  it  is  difficult  to  conceive  how  the  rays  of  light,  unless  they  be  bodies,  can  have  a  per- 
manent virtue  in  two  of  their  sides  which  is  not  in  their  other  sides,  and  this,  without  any  regard  to  their  position 
as  to  the  space  or  medium  through  which  they  pass." 

975.  Although  we  have  no  knowledge  of  the  intimate  constitution  of  elastic  media,  or  the  manner  in  which  their 
Examined,  contiguous  particles  are  related  to  each  other  and  affect  each  other's  motion,  yet  it  is  certain  that  the  mode  and 
laws  of  the  propagation  of  motion  through  them  by  undulation  cannot  but  depend  very  materially  on  this  con- 
nection. The  only  analogies  we  have  to  guide  us  into  any  inquiry  into  these  laws,  are  those  of  the  propagation 
of  sound  in  air  or  water,  and  of  tremors  through  elastic  solids,  and  along  tended  chords  and  surfaces ;  and  such  is 
the  extreme  difficulty  of  the  subject  when  taken  up  in  a  purely  mathematical  point  of  view,  that  we  are  forced  to 
have  recourse  to  these  analogies,  and,  dismissing  in  the  present  state  of  science  the  vain  hope  of  embracing  the 
whole  subje.ct  in  analytical  formulae,  suffer  ourselves  to  be  instructed  by  experience,  as  to  what  modifications  the 
peculiar  constitution  of  vibrating  media  may  produce  in  the  propagation  of  motion  through  them.  Now,  when 
sound  is  propagated  through  air  or  water,  in  which  the  molecules  are  at  least  supposed  to  have  no  mutual  con- 
nection but  to  be  capable  of  moving  with  equal  facility,  and  to  be  restored  to  their  places  with  equal  elastic 
forces,  in  whatever  direction  they  are  displaced,  and  in  which,  moreover,  it  is  (at  least  theoretically)  taken  for 
granted,  that  the  motion  of  any  molecule  has  an  equal  tendency  to  set  in  motion  those  adjacent  to  it,  in  what- 
ever direction  these  may  be  situated  with  respect  to  it;  it  is  difficult  to  conceive  that  the  motion  of  a  molecule  in 
the  surface  of  a  wave,  at  some  distance  from  the  centre  whence  the  sound  emanates,  can  be  performed  otherwise 
than  in  the  direction  of  the  radius,  or  at  right  angles  to  the  surface  of  the  wave ;  so  that  in  this  case  the  motion 
of  the  vibrating  molecules  must  coincide  with  the  direction  of  the  rays  of  sound,  and  there  appears,  therefore,  no 
reason  why  such  rays  should  bear  different  relations  to  the  different  regions  of  space  surrounding  them,  whether 
right  or  left,  above  or  below;  for  the  ray  being  regarded  as  an  axis,  all  parts  of  the  sphere  round  it  are  similarly 
related  to  it. 

976-  But  if  we  conceive  a  connection  of  any  kind,  such  as  may  possibly  be  established  by  repulsive  and  attractive 
forces,  or  magnetic  or  other  polarities  subsisting  between  the  molecules  of  the  vibrating  medium,  the  case  is 
altered.  It  will  no  longer  then  follow  of  necessity,  that  the  individual  motion  of  each  molecule  is  performed  in 
the  direction  in  which  the  general  wave  advances,  but  it  may  be  conceived  to  form  any  angle  with  that  direction, 
even  a  right  angle.  A  familiar  instance  of  such  a  mode  of  propagation  may  be  seen  in  the  wave  which  runs  along 
along  stretched  cord,  struck,  shaken,.or  otherwise  disturbed  at  one  end.  The  direction  of  the  wave  is  the  length 
of  the  cord,  and  that  of  the  motion  of  each  molecule  lies  in  a  plane  perpendicular  to  it.  Now  this  is  precisely  the 
Fresnel's  kind  of  propagation  which  M.  Fresnel  conceives  to  obtain  in  the  case  of  light.  He  supposes  the  eye  to  be 
affected  only  by  such  vibrating  motions  of  the  ethereal  molecules  as  are  performed  in  planes  perpendicular  to  the 
vibrations,  directions  of  the  rays.  According  to  this  doctrine,  a  polarized  ray  is  one  in  which  the  vibration  is  constantly 
performed  in  one  plane,  owing  either  to  a  regular  motion  originally  impressed  on  the  luminous  molecule,  or  to 
some  subsequent  cause  acting  on  the  waves  themselves,  which  disposes  the  planes  of  vibration  of  their  mole- 
cules all  one  way.  An  unpolarized  ray  may  be  regarded  as  one  in  which  the  plane  of  vibration  is  per- 
petually varying,  or  in  which  the  vibrating  molecules  of  the  luminary  are  perpetually  shifting  their  planes  of 
motion,  and  in  which  no  cause  has  subsequently  acted  to  bring  the  vibrations  thus  excited  in  the  ether  to 
coincident  planes. 

977.          The  analogy  of  the  tended  cord  (which  appears  to  have  suggested  itself  to  Dr.  Young  on  considering  the 
Propagation  optical  properties  of  biaxal  crystals  in  1818)  will  help  our  conception  greatly.     Suppose  such  a  cord  of  indefinite 
length,  stretched  horizontally,  and  one  end  of  it  being  held  in  the  hand,  let  it  be  agitated  to  and  fro  with  a 
s-rf     mot'on  perpendicular  to  the  length  of  the  cord.     Then  will  a  wave  or  succession  of  waves  be  propagated  along  it, 
waves  along  an^  every  molecule  of  the  cord  will,  after  the  lapse  of  a  time  proportional  to  its  distance  from  the  hand,  begin 
a  stretched   to  describe  a  line  or  curve  similar  and  similarly  situated  to  that  described  by  the  extremity  at  which  the  agitation 
eord.  originates.     If  the  original  agitation  be  regularly  repeated  and  constantly  confined  to  one  plane,  (he  same  will 

be  true  of  the  motion  of  each  molecule,  and  the  whole  extent  of  the  cord  will  be  thrown  into  the  form  of  an  undu- 
".ting  curve  lying  in  one  plane,  so  far  as  the  motion  has  reached.  In  this  case  it  will  represent  a  polarized  ray 
or  system  of  waves.  If,  after  a  few  vibrations  in  one  plane,  the  extremity  be  made  to  execute  a  few  in  another, 
and  then  again  in  another,  and  so  on,  so  that  the  plane  of  vibration  shall  assume  in  rapid  succession  all  pos- 
sible situations,  since  each  molecule  obeys  exactly  the  same  law  of  motion  with  the  extremity,  the  curve  will 
consist  of  portions  lying  in  all  possible  planes,  and  since  by  reason  of  the  propagation  of  the  undulation  along  it, 
every  point  of  it  is  in  succession  agitated  by  the  motion  of  every  other,  all  these  varied  vibrations  will  run 
through  any  given  point  of  it,  and  were  a  sentient  organ  like  the  human  retina  stationed  there,  the  impression 
it  would  receive  would  be  analogous  to  that  excited  in  the  eye  by  an  unpolarized  ray  of  light. 

•i'8.  It  may  be  objected  to  this  mode  of  conceiving  the  luminiferous  undulations,  that  the  molecules  of  the  ether,  if 

Ohjectiins    it  be  a  fluid,  such  as  we  have  hitherto  all  along  regarded  it,  cannot  be  supposed  connected  in  strings,  or  chains 
considered,   like  those  of  a  tended  cord,  but  must  exist  separate  and  independent  of  each  other.    But  it  is  sufficient  for  our  pur- 
pose to  admit  such  a  degree  of  lateral  adhesion  (we  hesitate  to  term  it  viscosity')  as  may  enable  each  molecule  in 
its  motion  not  merely  to  push,  before  it  those  whi^K  lie  directly  in  the  line  of  its  motion,  but  to  drag  along 


LIGHT.  535 

Light,  with  it  those  which  lie  on  either  side,  in  the  same  direction  with  itself.  Or,  acknowledging  at  once  tne  1'art  IV. 
— V"™-"  difficulty,  since  light  is  a  real  phenomenon,  we  are  not  to  expect  it  to  be  produced  without  a  mechanism  ^^ v""*' 
adequate  to  so  wonderful  an  effect.  We  do  not  hesitate  to  attribute  to  the  fluids  which  are  imagined  to  account 
for  the  phenomena  of  heat,  electricity,  magnetism,  &c.  properties  altogether  repugnant  to  our  ordinary  notions 
of  fluids,  and  why  should  we  deny  ourselves  the  same  latitude  when  light  is  to  be  accounted  for.  It  is  true 
the  properties  we  must  attribute  to  the  ether  appear  characteristic  of  a  solid  than  of  a  fluid,  and  may  be 
regarded  as  reviving  the  antiquated  doctrine  of  a  plenum.  But  if  the  phenomena  can  be  thereby  accounted 
for,  z.  e.  reduced  to  uniform  and  general  principles,  we  see  no  reason  why  that,  or  any  still  wilder  doctrine, 
should  not  be  admitted,  not  indeed  to  all  the  privileges  of  a  demonstrated  fact,  but  to  those  of  its  represen- 
tative, or  locum  tenens,  till  the  real  truth  shall  be  discovered.  Assuming  it,  then,  with  M.  Fresnel,  as  a  pos- 
tulatum,  that  the  vibrations  of  the  ethereal  molecules  which  constitute  light  are  performed  in  planes  at  right 
angles  to  the  direction  of  the  ray's  progress,  let  us  see  what  account  can  be  given  of  the  phenomena  of 
polarized  light. 

And  first,  then,  of  the  interference  of  two  polarized  rays,  whether  polarized  in  the  same,  or  different  planes.       979. 
The  plane  of  polarization  in  this  doctrine  may  be  assumed  to  be  either  that  in  which  the  vibrations  are  executed,  Explanation 
(i.  e.  a  plane  passing  through  the  direction  of  the  ray  and  the  line  described  by  each  of  the  vibrating  molecules  °^* ^f" 
in  its  excursion,)  or  one  perpendicular  to  it,  which  we  please.     Reasons,  presently  to  be  stated,  render  the  latter  interference 
preferable,  but  at  present  it  is  a  matter  of  indifference  which  we  assume.     Now,  in  §  3,  Part  III.  we  have  on  this 
investigated  at  length,  with  a  view  to  the  present  inquiry,  the  modes  of  vibration  which  result  from  the  combi-  doctrino. 
nation  of  any  assigned  vibrations,  whether  executed  in  the  same  or  different  planes ;   and  it  follows  from  the 
purely  mechanical  principles  there  laid  down,  1st,  That  the  combination  of  two  vibrations  executed  in  the  same 
plane,  produces  a  resultant  vibration  in  the  same  plane,  which  may  be  of  any  degree  of  intensity  from  the  sum 
to  the  difference  of  the  intensities  of  its  component  vibrations,  according  to  the  difference  of  their  phases.    Now, 
each  of  these  systems  of  vibration  represents  a  polarized  ray ;   so  that  rays  polarized  in  the  same  plane  ought, 
on  these  principles,  to  be   capable  of  destroying  or  reinforcing  each  other  by  interference,  as  we  see  they  do. 
But  the  case  is  otherwise  when  the  component  vibrations  are  executed  in  different  planes,  for  in  that  case  it  is 
obvious  that  they  never  can  destroy  each  other  completely  so   as  to  produce  rest.     The  general  case  of  non- 
coincident  planes  of  vibration  is  analyzed  in  Art.  618;  and  in  Art.  621  we  see,  that  even  when  each  of  the 
component  vibrations  is   rectilinear,   the  resultant  is  elliptic  ;    so  that  each  molecule  of  the  ether  performs 
continual  gyrations  in  one  direction,  and  never  can  be  totally  quiescent. 

Thus  we  see  that  the  interference  of  rays  similarly  polarized,  and  the  non-interference  of  those  dissimilarly,       9KO. 
is  a  necessary  consequence  of  the  hypothesis  we  are  considering;  and  indeed  was  the  phenomenon  which  first  Analogy 
suggested  it.     It  may  be  familiarly  explained  by  the  analogy  of  our  tended  cord.     Conceive  such  a  cord  to  of  the 
have  its  extremity  agitated  at  equal  regular  intervals  with  a  vibratory  motion  performed  in  one  plane,  then  it  *^ 
will  be  thrown,  as  we  have  seen,  into  an  undulatory  curve,  all  lying  in  the  same  plane.     Now,  if  we  superadd 
to  this  motion  another,  similar  and  equal,  but  commencing  exactly  half  an  undulation  later,  it  is  evident  that  the 
direct  motion  every  molecule  would  assume,  in  consequence  of  the  first  system,  will  at  every  instant  be  exactly 
neutralized  by  the  retrograde  motion  it  would  take  in  virtue  of  the  other;    and,  therefore,  each  molecule  will 
remain  at  rest,  and  the  cord  itself  be  quiescent.     But  if  the  second  system  of  motions  be  performed  in  a  plane 
at  right  angles  to  the  first,  the  effect  will  evidently  only  be  to  distort  the  figure  of  the  cord  into  a  curve  of  double 
curvature,  which,  in  the  general  case,  will  be  an  elliptic  helix,  and  will  pass  into  the  ordinary  circular  one  when 
the  two  component  vibrations  differ  in  phase  by  a  quarter  of  an  undulation,  or  90°.    (See  Art.  627.  Carol.) 

In  this  case  the  extremity  of  the  cord  describes  a  circle  with  a  continuous  motion,  and  this  motion  is  imi-        981. 
tated  by  each  molecule  along  its  whole  length.     It  is  easy  to  make  this  a  matter  of  experiment;  we  have  only  Case  of  a 
to  hold  in  our  hands  the  end  of  a  long  stretched  cord,  or  grasp  it  firmly  in  any  part  of  its  extent,  and  work  the  r°|alory  ": 
part  held  round  and  round,  with  a  regular  circular  motion,  and  we  shall  see  the  cord  thrown  into  a  helicoidal  ^otion 
curve,  each  portion  of  which  circulates  in  imitation  of  the  original  source  of  the  motion 

But  experience  shows,  not  merely  that  two  equal  rays  polarized  at  right  angles  do  not  destroy  each  other  for       982 
any  assignable  difference  of  origins,  but,  that  whatever  be  this  difference,  the  intensity  of  the  resultant  rayremains  Resultant  o( 
absolutely  the  same.     Now  this  is  also  a  necessary  consequence  of  the  theory  of  transverse  vibrations.     To  show  two  rays 
this,  we  need  only  refer  to  the  expressions  for  A,  B,  C  in  equation  (7,)  Art.  619,  resuming  at  the  same  time  the  °^fa0r^y 
notation  and  reasoning  of  that  article.     The  intensity  of  the  impression  made  on  the  eye  by  any  ray  being  investigated 
proportional  to  the  vis  viva,  is  represented  by  the  sum  of  the  several  vires  viva  in   the  three  rectangular 
directions,  or  by  A2  -f-  B-  -f-  C-,  that  is,  by 

aa  +  62  +  c2  -f  a'*  +  &'«  +  c'2  +  2  a  a' .  cos  (p  -  j/)  +  2  b  b' .  cos  (q  -  q')  -f-  2  c  c  .  cos  (r  -  /). 

Now  if  we  assume  the  directions  of  the  coordinates  x  and  y  to  be  those  transverse  to  that  of  the  ray,  and  the 
one  in  the  plane  of  polarization  of  one  ray,  the  other  in  that  of  the  other,  at  right  angles  to  it,  and  that  of  z 
in  the  direction  of  the  ray  itself,  we  have 

a!  =  0,     6=0      c  =  0,     (f  =  0  ; 
and  therefore  the  above  expression  for  the  intensity  becomes 

A«       B2  -f  C*  =  a"-  +  6'2, 
whicn  is  independent  of  p  •—  p',  tj  —  q\  r  —  r,  the  difference  of  phases,  and  is  equal  to  tne  sum  ot  the  mten- 


536  LIGHT. 

Light,     sities  of  the  separate  rays.     And  we  may  remark,  by  the  way,  that  no  other  supposable  mode  of  vibration  but     Part 
— •"V™"'  that  in  question,  in  which  c  and  </,  the  amplitudes  of  vibration  in  the  direction  of  the  ray  vanish,  could  produce     "~v 
the  same  result.     (Fresnel's  Considerations  Theoriqites  siir  la  Polarization  de  la  Lumiere.      Bulletin  de  In 
Socitte  Philomatiqite,  October,  1924.) 

983.  Let  us  now  consider  what  will  happen  when  a  ray  polarized  in  any  plane  is  resolved  into  two  polarized 

Rationale  of  in  any  other  two  planes  at  right  angles  to  each  other,  and  these  again  reduced  to  two  others  also  at  right 
"angles  to  each  other,  by  a  second  resolution.     Suppose  C,  (fig.  195,  a)  to  be  the  course  of  a  ray  projected  on  a 
]°undula-    P'ane  perpendicular  to  its  direction,  (that  of  the  paper,)  and  in  which,  consequently,  the  vibrations  of  the 
tion.  molecule  C  are  performed.     Let  P  C  P7  be  the  line  of  vibration  of  this  molecule,  and  therefore  (according  to  the 

Fig.  195.  hypothesis  assumed)  at  right  angles  to  the  plane  of  primitive  polarization.  When  this  ray  is  divided  into  two 
others  oppositely  polarized,  the  vibrations  are  of  course  resolved  into  two  others  performed  in  planes  at  right 
angles  to  each  other.  Let  C  O  and  C  E  be  the  projections  of  these  planes,  which  are  therefore  perpendicular 
to  the  planes  of  polarization  of  the  two  new  rays  respectively.  Suppose  that  at  any  instant  the  molecule  C  of 
the  primitire  ray  is  moving  from  C  in  the  direction  C  P ;  then  this  motion,  if  resolved  into  two,  will  give  rise 
to  two  motions,  one  in  the  direction  from  C  towards  O,  the  other  from  C  towards  E.  If  each  of  these  motions 
be  again  resolved  into  two,  in  planes  whose  projections  are  S  C  S'  and  T  CT',  at  right  angles  to  each  other,  that 
in  the  direction  C  O  will  produce  two  motions,  one  in  the  direction  C  S,  and  the  other  in  the  direction  C  T ; 
and  on  the  other  hand  the  motion  in  the  direction  C  E  will  produce  one  in  the  direction  C  S,  and  the  other  (in 
the  case  of  fig.  195,  a)  in  the  direction  C  T'  opposite  to  C  T.  Thus  the  two  resolved  motions  in  the  plane  S  S' 
will  conspire,  but  those  in  the  plane  TT'  will  oppose,  each  other.  In  the  case  of  fig.  195,  6,  the  reverse  will 
happen  ;  the  motions  in  the  plane  T,  1"  conspiring,  and  those  in  the  plane  S  S'  opposing,  each  other.  For  sim- 
plicity of  conception,  however,  we  will  confine  ourselves  to  the  former  case.  If,  now,  we  pass  from  the  consi- 
deration of  the  vibrations  to  that  of  the  rays,  it  will  appear  that  we  have,  in  fact,  resolved  the  original  ray 
polarized  in  the  plane  P  P'  into  two,  polarized  in  planes  perpendicular  respectively  to  C  O  and  C  E  ;  and  these 
again,  finally,  each  into  two,  viz.  one  polarized  in  the  perpendicular  to  S  S',  and  one  perpendicular  to  TT*. 
The  two  portions  polarized  perpendicular  to  S  S'  form  one  ray,  and  those  perpendicular  to  TT'  another;  but  in 
the  former,  the  component  portions  tend  to  strengthen, — in  the  latter,  to  destroy  each  other.  Hence,  if  we 
consider  the  two  former  portions  as  having  a  common  origin,  we  must  regard  the  latter  as  differing  by  hall 
an  undulation. 

984.  Hitherto  we  have  supposed  the  second  resolution  of  the  rays  to  take  place  at  the  same  point  C  in  the  course 
of  the  ray  as  the  first,  but  this  may  not  be  the  case,  and  several  cases  may  be  imagined ;  first,  we  may  suppose 
the  two  portions  into  which  the  ray  is  first  resolved  to  run  on  in  the  same  line  with  equal  velocities  ;  and  after 
describing  any  given  space,  to  be  then  resolved,  at  another  point  C'  (whose  projection  in  the  figure  will  coincide 
with  C)  into  the  final   rays  S  S'  and  T  T'.     It  is  evident  that  this  will  make  no  difference  in  the  result,  for  the 
phases  in  which  each  ray  arrives  at  C'  will  be  alike  ;  and  after  the  second  resolution  the  conspiring  vibrations 
in    the  direction  S  S'  will  still  be  in  the  same  phase,  and  the  opposing  ones  in  the  plane  TT'  must  still  be 
regarded  as  in  opposite  phases,  i.  e.  as  differing  by  half  an  undulation.     Or,  secondly,  we  may  suppose,  that, 
owing  to  any  cause,  the  two  resolved  rays  do  not  travel  with  equal  velocity,  (as  in  the  case  where  the  reso- 
lution is  performed  by  double  refraction.)     In  this  case,  if  i  be  the  interval  of  retardation  of  the  one  ray  on 
the  other  when  they  arrive  at  C',  i  will  represent  the  difference  of  phases  of  the  two  rays  at  the  instant  of  their 
second  resolution.     Consequently,  when  resolved,  the  final  ray,  whose  vibrations  are  performed  in  S  S',  will  be 
the  sum ;  and  that  whose  vibrations  are  performed  in  TT',  the  difference  of  two  rays,  one  in  a  certain  phase  (6), 
the  other  in  the  phase  0  -f-  i  ;   or,  which  is  the    same  thing,  the  former  will   be  the  sum  of  two  components 
in  the  phases  0  and  0  -\-  i ', ;  the  latter,  the  sum  of  two  in  the  phases   0  and  0  -j-  i  -}-  180°,  so  that  still  the 
difference  of  half  an  undulation  is  to  be  applied.     In  the  case  of  fig.  195,  b,  if  we  pursue  the  same  reasoning, 
it  will  appear  that  this  difference  still  subsists,  but  must  be  applied  conversely,  viz.  to  the  compound  ray  whose 
vibrations  are  performed  in  C  S. 

985.  We  have  here,  then,  the  theoretical  origin  of  the  allowance  of  half  an  undulation,  in  those  cases  where  it  is 
required  to  account  for  the  polarized  tints,  Art.  966,  and  of  the  rule  laid  down  in  Art.  972  for  its  correct  appli- 
cation.    However  arbitrary  the  assumption  may  have  appeared  as  there   presented,  and  however  singular  it 
may  have  seemed  to  make  the  affections  of  a  ray  at  one  point  of  its  course  dependent  on  those  which  it  had 
at  a  former  instant,  we  now  see  that  the  whole  is  a  direct  and  very  simple  consequence  of  the  ordinary  elemen- 
tary rules  for  the  composition  and  resolution  of  motions.     It  is  worthy  of  notice,  that  the  fact  was  ascertained 
before  .the  theory  of  transverse  vibrations  was  devised,  so  that  this  theory  has  the  merit  of  affording  an  a  priori 
explanation  of  what  had  previously  all  the  appearance  of  a  mere  gratuitous  hypothesis. 

986.  In  conceiving  the  resolution  of  a  ray  into  two  others  polarized  in  different  planes,  we  may  be  aided  by  the 
Application  analogy  of  the  tended  cord,  which  we  have  before  had  occasion  to  refer  to.     In  fig.  196  let  A  B  be  a  stretched 
"'  the  al)a-  cord,  branching  at  B  into  the  two  B  C  arid  B  D,  making  a  small  angle  with  each  other  at  B,  and  having  either 
stretched  *  eclual  or  unequal  tensions.     Suppose  the  plane  in  which  the  two  branches  lie  to  be  (for  illustration's  sake)  hori- 
cord.  zontal,  and  let  the  extremity  A  of  the  single  cord  be  made  to  vibrate  regularly  in  a  vertical  plane ;   or,  at  least 
Fig.  196.  let  the  vibrations  of  the  cord,  before  arriving  at  B,  be  reduced  to  a  vertical  plane  by  means  of  a  small  polished 

vertical  guide  I  K,  against  which  the  cord  shall  press  lightly,  and  on  which  it  may  slide  freely  without  friction. 
Beyond  the  point  of  bifurcation  B,  and  at  such  a  distance  that  the  excursions  of  the  molecule  B  shnll  subtend 
no  sensible  angle  from  them,  let  two  other  such  polished  guiding  planes  be  placed,  inclined  at  different  angles 
to  the  horizon,  and  making  a  right  angle  with  each  other.  Suppose  now  B  to  make  any  excursion  from  its 
point  of  rest,  then  were  the  plane  E  F  parallel  to  I  K,  the  molecule  of  the  branch  B  C  contiguous  to  E  F  would 
slide  on  E  F  through  a  space  equal  to  the  whole  excursion  of  B ;  but  since  it  is  inclined  to  I  K  at  an  angle 


LIGHT.  537 

(=  G)  a  part  only  of  the  motion  of  B  will  be  employed  in  causing  this  molecule  to  glide  on  E  F,  and  the  P^t  IV. 
remainder  will  cause  the  cord  to  bend  over  and  press  on  the  obstacle ;  but  by  reason  of  the  minuteness  of  the  v— •" v~™ * 
excursions  of  B,  this  bending  and  the  resistance  of  the  obstacle  and  consequent  loss  of  force  will  be  very  minute 
and  may  be  neglected.  Now,  since  the  pressure  of  the  obstacle  removes  the  cord  from  the  position  it  would 
have  taken  had  no  obstacle  existed,  in  a  direction  perpendicular  to  its  surface,  it  is  easy  to  see  that  the 
amplitude  of  excursion  of  the  contiguous  molecule  on  the  plane  E  F  must  be  to  that  of  B  as  cos  0  to  radius  ; 
and,  therefore,  calling  a  the  amplitude  of  B's  excursions,  a  .  cos  0  will  be  that  of  the  molecule  contiguous 
to  E  F,  and  of  course  that  of  every  subsequent  molecule  of  the  branch  B  C.  Here  the  part  of  B's  motion, 
which  is  perpendicular  to  E  F,  is  not  expended  or  destroyed  in  bending  the  cord  B  C  over  the  obstacle,  but 
remains  in  activity,  and  exerts  itself  on  the  branch  B  D,  causing  it  to  glide  on  the  plane  G  H  ;  and  the  ampli- 
tude of  the  excursions  of  the  molecule  in  contact  with  this  plane  will  in  like  manner  be  represented  by  a  .  cos 
(inclination  of  G  H  to  I  K,)  that  is,  by  a  .  cos  (90  —  0),  or  by  a  .  sin  0.  The  vis  viva,  then,  in  each  of  these 
respective  planes  is  represented  by  a',  cos  6*  and  a1 .  sin  6*,  whose  sum  is  equal  to  a"1,  the  initial  vis  viva. 

If  we  decompose,  in  like  manner,  the  maximum  velocity  «  of  the  ethereal  molecule  C  (fig.  195)   in  the       987. 
direction  C  P  into  two  in  the  respective  directions  C  O  and  C  E,  we  get  a  .  cos  0  and  a  .  sin  0  for  the  elementary  Rationale  of 
velocities;    and  since  the  amplitudes,  cceteris  paribus,  are  as  the  velocities,   (Art.  filO,)  the  amplitudes  of  the  Malu?'s . , 
component  rays  will  be  respectively  a .  cos  0  and  a  .  sin  0  ;  and  their  intensities,  which  are  as  the  squares  of  the  jnt^j|_tj| 
amplitudes,  (Art.  fi05,)  will  be  a" .  cos  0s  and  a1 .  sin  0*.     Now  this  is  the  very  law  propounded  by  Mains  for  the  ti,e  comple- 
intensities  of  the  two  portions  into  which  a  polarized  ray  is  divided  by  double  refraction,  and  of  which  the  mentary 
theory  of  transverse  vibrations  gives,  as  we  see,  a  simple  and  rational  a  priori  account,  thus  raising  it  from  raJ's- 
a  mere  empirical  law  to  the  rank  of  a  legitimate  theoretical  deduction. 

We  have  not  done  with  the  analogy  of  the  tended  cord.     What  we  have  shown  in  Art.  986  is  independent  of      988. 
the  tensions  of  the  branches  into  which  the  cord  is  divided,  and  relates  only  to  the  amplitudes  of  their  excur-  Case  of  the 
sions  from  rest  when   thrown  into  vibration.     But  the  velocity  with  which  the  waves,  once  produced,  will  be  twn  r'r~ 
propagated  along  either  branch  depends  solely  on  its  tension.     Nothing,  however,  prevents  the  tensions  of  the  delations"" 
two  branches  from  being  very  different ;  for,  whatever  be  the  ratio  of  two  forces  applied  in  the  directions   B  C  propagated 
and  B  D,  they  may  be  balanced  at  B  by  a  proper  force  applied  along  any  other  line  as  B  A.     Hence  the  waves  with 
will  run  along  B  C  and  B  D  with  different  velocities.     Similarly,  if  we  conceive,  that  owing  to  the  peculiar  different 
constitution  of  crystallized  bodies,  and  the  relation  of  their  particles  to  the  ether  which  pervades  them,  its  mole-  ve 
cules  are  more  easily  displaced,  or  yield  to  a  less  force  in  certain  planes  than  in  others ;  or,  in  other  words, 
that  it  possesses  different  elasticities  in  different  directions  ;    then  will  the  planes  of  polarization  assumed  by 
the  resolved  portions  of  the   rays  determine  the  elasticities  brought  into  action,  and,  by  consequence,    the 
velocities  of  their  propagation.     Now  we  have,  in  a  former  section,  shown  that  the  bending  of  a  ray  at  the. 
confines  of  a  medium  depends  essentially  on  its  velocity  within  as  compared  with  that  without,  by  the  analytical 
relations  deduced  from  the  "  principle  of  swiftest  propagation."     A  difference  of  velocity,  therefore,  draws  with 
it,  as  a  necessary  consequence,  a  diversity  of  path  ;    and  thus  the  bifurcation,  or  double  refraction  of  a  ray 
incident    on  a  crystallized  surface,    presents  no    longer    any  difficulty   in  theory,  provided    we    can    find    an 
adequate  reason  for  the  resolution  of  its  vibrations  into  two  determinate  planes  at  the  moment  of  its  entering 
the  crystal.  t 

Let  us  take  (with  M.  Fresnel,  Annales  de  Chimie,  xvii.   p.  179  et  scq.)  the  case  of  a  crystal  with  one  axis.       98^- 
We  may  regard  this,  or  rather  the  ether  within  it,  modified  in  its  action  by  the  molecular  forces  of  the  crystal,  Explanation 
as  an  elastic  medium  in  which  the  elasticity  in  a  direction  perpendicular  to  the  axis  is  different  from  that  in  a  n0menaPof 
direction  parallel  to  it,  that  is,  in  which  the  molecules  are  more  easily  compressible  in  the  one  than  in  the  other  double 
direction  ;  but,  equally  so  in  all  directions  perpendicularly  to  the  axis,  on  whatever  side  the  pressure  be  applied,  refraction 
To  aid  our  conceptions  in   imagining  such  a  property,  we  may  assimilate  an  uniformly  elastic  medium  to  an  in_  crystals 
assemblage  of  thin,  elastic,  hollow,  spherical  shells  in  contact;  and  such  a  medium  as  we  are  considering,  to  a  wl 
similar  assemblage  of  oblate  or  prolate  hollow  ellipsoids,  arranged  with  all  their  axes  parallel  to  one  common  direc- 
tion,  which  is  that  of  the  axis  of  the  crystal.*    It  is  evident  that  the  resistance  of  the  (spherical  assemblage  to  pressure 
must  be  the  same  in  all  directions,  but  that  of  the  spheroidal  must  differ  according  as  the  pressure   is  applied 
perpendicularly  or  parallel  to  the  axis.     Thus,  it  is  easy  to  crush  an  egg  by  a  force  applied  in  the  direction 
of  its  shorter  diameter,  which  will  yet  sustain  a  violent  pressure  applied  at  the  extremities  of  its  longer.     It  is, 
moreover,  evident,  if  any  molecule  of  such  an  assemblage  were  disturbed,  so  a?  to  throw  it  into  vibration,  that, 
provided  always  the  amplitude  of  its  excursions  were  extremely  small  compared  to  the  diameter  of  each  ellipsoid, 
the  immediate  tendency  of  the  vibration  will  be  to  communicate  motion  to  two  strata  only  of  molecules,  viz.  that 
in  which  the  axis  and  equator  of  the  disturbed  molecule  lie  respectively,  since  it  is  only  at  the  poles  and  equator 
that  they  touch,  and  therefore  only  through  these  points  that  motion   can  be  communicated  from  one  to  the 
other.     Consequently,  any  motion  communicated   to  a  molecule  of  such  a  mass  could  only  be  propagated  by 
vibrations  performed  in   planes  parallel  and  perpendicular  to  the  axis.     Hence,  if  a  vibratory  motion  in  any 
plane  be  propagated  into  such  aa  assemblage  of  particles  from  without,  it  will  immediately,  on  its  reaching  it. 


VOL,  IV 


538  LIGHT. 

l.ib'ht.      he  resolved  into  two,  in  the  planes  above  named ;    and  these,  by  reason  of  the  different  elasticities,  will  be     Part  IV 
*— — v— '  propagated  with  different  velocities. 

990.  The  reader  must  not  suppose  that  this  is  intended  for  an  account  of  the  real  mechanism  of  crystallized  bodies. 
Bifurcation    jt  jg  merely  intended  to  show  that  it  is  not  absurd,  or  contradictory  to   sound  mechanical  principles,  to  assume 
fracted'ra     t^lat  suc'1  ma^  ^e  ^'^  const'tut'on'  tnat  vibrations  can  only  be  propagated  through  them  by  molecular  excur- 
explained      sions  executed  in  pjanes  parallel  and  perpendicular  to  their  axes.     Assuming,  then,  that  such  is  the  case,  the 

vibrations  of  a  ray  incident  on  such  a  crystal  will  be  resolved  into  two,  performed  in  these  respective  planes,  and 

their  velocities  of  propagation  being  different,  the  rays  so  arising  will  follow  different  courses  when  bent  by 

refraction.     Let  us  first  consider  that  whose  vibrations  are  executed  in  planes  perpendicular  to  the  axis.     Since 

the  crystal  is  symmetrical  with  respect  to  i«s  axis,  and  equally  elastic  in  all  directions  perpendicular  to  it,  the 

Properties    velocity  of  propagation  of  this  portion  will  be  the  same  in  all  directions.    Its  index  of  refraction,  therefore,  will  be 

oftheordi-   constant,  and  the  refraction  of  this  portion  will  follow  the  ordinary  law.     Moreover,  its  plane  of  polarization 

nary  ray.      being  that  perpendicular  to  which  the  vibrations  are  performed,  will  necessarily  pass  through  the  axis,  in  which 

respect  it  also  agrees  with  the  ordinary  ray,  as  actually  observed. 

991.  The  extraordinary  ray  arises  from  the  other  resolved  portion  of  the  original  vibration,  which  is  performed  in  a 
Properties     plane  parallel  to  the  axis.      By   the  principle  of  transverse  vibrations,  it  is  also  performed  in  a  plane  per- 
oftheextra-  peiidicular  to  the  ray.     If,  then,  we  suppose  a  plane  to  pass  through  the  extraordinary  ray  and  the  axis,  it  will 

'xd'laineday  cut  a  P'ane  perpendicular  to  the  ray  in  a  straight  line,  which  will  be  the  direction  of  the  vibratory  motion.  This 
direction,  then,  is  inclined  to  the  axis  in  an  angle  equal  to  the  complement  of  that  made  by  the  extraordinary 
ray  with  the  latter  line,  and  therefore,  when  the  extraordinary  ray  is  parallel  to  the  axis,  the  line  of  vibration  is 
perpendicular  to  it,  and  vice  versd.  In  the  former  case,  the  elastic  force  resisting  the  displacement  of  the  mole- 
cules is  the  same  as  in  the  case  of  the  ordinary  ray,  and  therefore  the  velocities  of  both  rays  are  equal,  and 
their  directions  coincide,  and  thus  along  the  axis  there  is  no  separation  of  the  rays.  In  the  latter,  the  elasticity 
is  that  parallel  to  the  axis,  and  therefore  differing  from  the  former  by  the  greatest  possible  quantity.  Here,  then, 
the  difference  of  velocities,  and  therefore  of  directions  is  at  its  maximum.  In  intermediate  situations  of  the 
extraordinary  ray,  the  elasticity  developed  is  intermediate,  and  therefore  also  the  velocity  and  double  refraction. 
Thus  we  see,  that  according  to  this  doctrine  the  difference  of  velocities,  and  consequent  separation  of  the  pencils 
should  be  nothing  in  the  axis,  and  go  on  increasing  till  the  extraordinary  ray  is  at  right  angles  to  it,  which  is 
conformable  to  fact.  Lastly,  the  plane  of  polarization  of  the  extraordinary  ray  being  at  right  angles  to  the 
plane  of  vibration,  must  also  be  at  right  angles  to  a  plane  passing  through  the  ray  and  the  axis,  which  is  also 
conformable  to  fact. 

992.  The  theory  of  M.  Fresnel  gives  then,  as  we  see,  at  least  a  plausible  account  of  the  phenomena  of  double 
refraction  in  the  case  of  uniaxal  crystals  ;  and  when  we  consider  the  profound  mystery  which,  on  every  other 
hypothesis,  was  admitted  to  hang  over  this  part  of  th«  subject,  we  must  allow  that  this  is  a  great  and  impor- 
tant step.     But  the  same  principles  are  equally  applicable  to  biaxal  crystals  with  proper  modifications,  and 
(which  is  a  strong  argument  for  their  reality)  lead,  when  so  applied,  to  conclusions  which,  though  totally  at 
variance  with  all  that  had  been  taken  for  granted  before,  on  the  grounds  of  imperfect  analogy  and  insufficient 
experiment,    have    been    since    verified    by   accurate  and  careful  experiments,  and  have  thus    opened  a  new 
and  curious  field  of  optical   inquiry      Nothing  stronger  can  be  said  in^  favour  of  an  hypothesis,  than  that  it 
enables  us  to  anticipate  the  results  of  experiment,  and  to  predict  facts  opposed  to  received  notions,  and  mis- 
taken or  imperfect  experience. 

993  But  before  we  enter  on  this,  it  may  be  right  to  show  how  the  phenomenon  on  which  the  theory  of  movable 

Explana-      polarization  is  founded,  is  accounted  for  by  the  doctrine  of  transverse  vibrations.     According  to  this  theory,  as 
tion  of  the    soon  as  a  polarized  ray  enters  a  crystal,  it  commences  a  series  of  alternate  assumptions  of  one  or  other  of 
phenomena   tvvo  planes  of  polarization,  in  the  azimuths  0°  and  2  i,  i  being  the  inclination  of  the  principal  section  to  the 
"olariza'tJon  P'ane  °^  primitive  polarization :  the  plane  assumed  being  in  azimuth  0°,  when  the  thickness  traversed  is  such 
"'  as  to  render  the  interval  of  retardation  of  the  ordinary  on  the  extraordinary  ray  0,  or  any  whole  number  ot 
undulations,   and  in  azimuth  2  i  when  it    is  any  whole  odd  number  of   semi-undulations.      Suppose  a  ray 
polarized  in  the  azimuth  0  to  be  incident  perpendicularly  on  a  crystallized  lamina,  having  its  principal  sec- 
tion in  the  azimuth  i,  then  it  will  be  resolved  into  two,  the  vibrations  of  which  are  respectively  performed  in  the 
principal  section,  and  perpendicular  to  it.     Consequently,  if  we  represent  by  unity  the  amplitude  of  the  original 
Case  of        vibrations,  those  of  the  two  resolved  vibrations  will  be  equal  respectively  to  sin  i  and  cos  i.     Now,  the  thick- 
complete      ness  of  the  plate  being  first  supposed  such  as  to  render  the  interval  of  retardation  an  exact  number  of  undula- 
accordance.  tions,  these  rays  will  emerge  from  the  lamina  in  exact  accordance,  and  being  parallel,  the  systems  of  waves  of 
which  they  consist  will  run  on  together.     Being  polarized,  however,  in  opposite  planes  they  will  neither  destroy 
each  other,  nor  produce  a  compound  ray  equal   to  their  sum,  but  their  resultant  must  be  determined  as  in 
Art.  623.     For  we  have  here  the  case  of  rectilinear  vibrations,  in  complete  accordance,  of  given   amplitudes, 
and  making  a  given  angle  (90°,)  so  that  the  result  there  obtained  is  immediately  applicable  to  this  case,  and 
the  resultant  vibration  will  be,  first,  rectilinear,  so  that  the  compound  ray  will  appear  wholly  polarized  in  one 
plane:  and,  secondly,  its  amplitude  will  be,  both  in  quantity  and   direction,   the  diagonal  of  a  parallelogram 
whose  sides  are  the  amplitudes  of  the  component  vibrations.     Consequently,  it  will  be  identical  with   that  by 
whose  resolution  these  were  produced,  and  therefore  the  resultant,  or  emergent  compound  ray  will  be,  in  respect 
gg,j        both  of  its  polarization  and  intensity,  precisely  similar  to  the  original  incident  one. 

Fig  197'.  When  the  difference  of  paths  within  the  crystal  is  an  exact  odd  multiple  of  half  an  undulation,  the  waves  at 
Case  of  their  egress  from  the  posterior  surface  will  be  in  complete  discordance.  But  their  resultant  may  still  be 
complete  determined  by  the  same  rule,  regarding  either  of  the  rays  as  negative,  i.  e.  as  having  its  vibrations  executed 
discordance.  ;n  tjje  Opp0sjte  direction.  F-V  suppose  the  molecule  C  moving  in  the  direction  C  P,  with  the  velocity  C  P 


LIGHT.  539 

Light,     (fig.    197)  at  the  entry  of  the  ray,  then  the  resolved  velocities  in  the  planes    CO  and  CE  will  be  repre-      Part  IV. 
— v"*-'  sented  in  quantity  and  direction  by  C  O  and  C  E.     But  at  their  egress,  the  vibrations  in  the  direction  C  E  v-">v™ •* 
having  gained  or  lost  a  half  undulation  on  those  in  C  O,  if  C  O  represent  the  quantity  and  direction  of  motion 
of  the  molecule  C  in  that  plane,  C  E'  equal  and  opposite  to  C  E  will  represent  its  motion  in  the  other  plane, 
and  this,  combined  with  C  O  will  compose,  not  the  original  motion  C  P,  as  in  the  former  case,  but  C  Q,  making 
an  equal  angle  with  C  O  on  the  other  side.     The  resultant  ray,  then,  instead  of  being  polarized  in  the  plane 
of  the  incident  one,   (i.  e.  perpendicular  to  C  P)  will  be  polarized  in  a  plane  perpendicular  to  C  Q,  making 
an  angle  equal  to  P  C  Q  (=  2  P  C  O  =  2  i)  with  CO. 

When  the  difference  of  routes  is  neither  an  exact  number  of  whole,  or  half  undulations,  the  vibrations  of       995 
the  resultant  ray  (by  Art.  621)  will  no  longer  be  rectilinear,  but  elliptic  ;   and  in  the  particular  case  when  the 
interval  of  retardation  is  a  quarter  or  an  odd  number  of  quarter  undulations,  it  will  be  circular.     In  this  case, 
the  emergent  ray,  varying  its  plane  of  vibration  every  instant,  will  appear   wholly  depolarized,  so  as  to  give 
two  equal  images  by  double  refraction  in  all  positions  of  the  analysing  prism. 

These  several  consequences  may  be  rendered  strikingly  evident  by  a  delicate  and  curious  experiment  related       996. 
by  M.  Arago.     Let  a  polarized  pencil,  emanating  from  a  single  radiant  point,  be  incident  on  a  double  rhomboid  Experiment 
of  Iceland  spar,  composed  of  two  halves  of  one  and  the  same  rhomboid,  superposed  so  as  to  have  their  principal  these severi 
sections  at  right  angles  to  each  other.     Then  the  emergent  rays  will  emanate  as  if  from  two  points  (see  Art.  879)  cases  Of 
near  each  other,  and  polarized  in  opposite  planes.     Let  these  two  cones  of  rays  be  received  on  an  emeried  glass,  interference 
or  in  the  focus  of  an  eye-lens,  so  that  the  glass  or  field  of  view  shall  be  illuminated  at  once  by  the  light  of  both, 
which  being  oppositely  polarized  will  exhibit  no  fringes  or  coloured  phenomena,  but  merely  a  uniform  illumina- 
tion ;  and  let  all  the  light  but  that  which  falls  on  a  single  very  small  point  of  the  field  of  view  be  stopped  by  a 
plate  of  metal,  with  a  small  hole  in  it,  so  as  to  allow  of  examining  the  state  of  polarization  of  the  compound  ray 
illuminating  this  point,  separately  from  all  the  rest.     Then  it  will  be  seen,  on  analysing  its  light  by  a  tourmaline 
or  double  refracting  prism,  that,  when  the  spot  examined   is  distant  from  both  radiants  by  the  same  number  of 
undulations,  although  in  fact  composed  of  two  rays  oppositely  polarized,  (as  may  be  proved  by  stopping  one  of 
them,  and  examining  the  other  singly,)  yet  it  presents  the  phenomenon  of  a  ray  completely  polarized  in  one 
plane,  which  is  neither  that  of  the  one  or  the  other  of  its  component  rays,  but  the  original  plane  of  polarization  of 
the  incident  light.    Suppose  now,  by  a  fine  screw  we  shift  gradually  the  place  of  the  metal  plate  so  as  to  bring  the 
hole  a  little  to  one  or  the  other  side  of  its  former  place.     The  ray  which  illuminates  it  will  appear  to  lose  its  pola- 
rized character  as  the  motion  of  the  plate  proceeds,  and  at  length  will  offer  no  trace  of  polarization ;  continuing  the 
motion,  and  bringing  in  succession  other  points  of  the  field  of  view  under  examination,  the  light  which  passes 
through  the  hole  will  again  appear  polarized,  at  first  partially,  and  at  length  totally ;  not,  however,  as  before,  in  the 
plane  of  primitive  polarization,  but  in  a  plane  making  with  it  twice  the  angle  included  between  it  and  the  principal 
section  of  the  first  rhomboid,  and  so  on  alternately.     Thus  we  are  presented  with  the  singular  phenomenon  of  two 
rays  polarized  in  planes  at  right  angles,  which  produce  by  their  concourse  a  ray  either  wholly  polarized  in  one  or 
the  other  of  two  planes,  or  not  polarized  at  all,  according  to  the  difference  of  routes  of  the  rays  before  their  union. 

In  1821,  M.  Fresnel  presented  to  the  Academy  of  Sciences  of  Paris  a  Memoir,  containing  the  general  appli-       997. 
cation  of  the  principle  of  transverse  vibrations  to   the  phenomena  of  double  refraction  and  polarization  as  Fresnel's 
exhibited  in  biaxal  crystals,  which  was  read  in  November  of  that  year.     A  brief  report  on  the  experimental  El!neral 
parts  of  this  Memoir  by  the  Committee  of  the  Academy  appointed  to  examine  it,  about  half  a  dozen  pages,  was  ^J^0 
published  in  the  Annales  de  Chimie,  vol.  xx.  p.  337,  recommending  it  to  be  printed  as  speedily  as  possible  in  refraction, 
the  collection   of  the  M&noires  des   Savans   Etraiigers.     We  are  sorry   to  observe,  that  this  recommendation 
has  not  yet  been  acted  upon,  and  that  this  important  Memoir,  to  the  regret  and  disappointment  of  men  of  science 
throughout  Europe,   remains  yet  unpublished  ;  though  we  trust  (from   the  activity  recently  displayed  by  the 
Academy  in   the  publication  of  their  Memoirs  in  arrear)   this  will  not  long  continue  to  be  the  case.  *     An 
abstract  by  the  author  himself,  which  appeared   in  the  Bulletin  de  la  Societe  Philomatique  of  1822,  and  was 
subsequently  reprinted  in  the  Annales  de  Chimie,  1825,  enables  us,   however,  to  present  a   sketch,  though  an 
imperfect  one,  of  its  contents,  supplying  to  the  best  of  our  ability  the  demonstration  of  the  fundamental  pro- 
positions, and  reaping  a  melancholy  gratification  from  the  inadequate  tribute,  which,  in  thus  introducing  for 
the  first  time  to  the  English  reader  a  knowledge  of  these  profound  and  interesting  researches,  we  are  enabled 
to  pay  to   departed  merit.     His  saltern  accumulem  donis — et  fungar  inani  munere.     For  even  at  the  moment 
when  we  are  recording  his  discoveries,  their  author  has  been  snatched  from  science  in  the  midst  of  his  brilliant 
career  by  a  premature  death,  like  his  hardly  less  illustrious  contemporary,  Fraunhofer,  the  early  victim  of  a 
weakly  constitution  and  emaciated  frame,  unfit  receptacles  fur  minds  so  powerful  and  active. 

M.  Fresnel   assumes,  as  a  postulatum,  that  the  displacement  of  a  molecule  of  the  vibrating  medium  in  a       998. 
crystallized  body  (whether  that  medium  be  the  ether,  or  the  crystal  itself,   or  both   together,  in  virtue  of  some  General  ex- 
mutual  action  exercised  by  them  on  each  other,)  is  resisted  by  different  elastic  forces,  according  to  the  different  pression  for 
directions  in  which  the  displacement  takes  place.     Now  it  is  easy  to  conceive,  that  in  general  the  resultant  of  forCe*otCa 

•• medium  m- 

*  This  delay  has  been  productive  of  a  singular  consequence,  which  will  suffice  to  show  the  small  degree  of  publicity  which  labours,  even  vestigated. 
the  most  important,  can  acquire  by  the  circulation  of  such  notices  as  those  mentioned  in  the  text.  So  lately  as  December  1826,  the 
Imperial  Academy  of  Sciences  of  Petersburg  proposed  as  one  of  their  prize  questions  for  the  two  years  1827  and  1828,  the  following,  "  To 
deliver  the  optical  system  of  waves  from  all  the  objections  which,  have  (as  it  appears*)  with  justice,  been  urged  against  it,  and  to  apply  it  to 
the  polarization  and  double  refraction  of  light,"  In  the  programma  announcing  this  prize,  M.  Fresnel's  researches  on  the  subject  are  noi 
alluded  to  (though  his  Memoir  on  Diffraction  is  noticed,)  and  it  is  fair  to  conclude,  were  not  then  known  to  the  Academy.  Precisely  one 
month  before  the  publication  of  this  programma,  the  Royal  Society  of  London  awarded  their  Rumford  Medal  to  M.  Fresnel,  "  for  his  appli- 
cation of  the  undulatory  theory  to  the  phenomena  of  polarized  light,  and  for  his  important  experimental  researches  and  discoveries  in  physical 
optics."  Our  readers  will  be  gratified  to  know,  that  the  valuable  mark  of  this  high  distinction  reached  him  a  few  days  before  his  death. 

4  A  2 


540  LIGHT. 

L.ght.      all  the  molecular  forces  which  act  on  a  displaced  molecule,  is  not  necessarily  parallel  to  the  direction  of  its  dis-     Part  IV. 
v—  -^^-  placements  when  the  partial   forces  are  unsymmetricully  related  to  this  direction,  but  the  proposition   may  be  v—  ~v-» 
demonstrated  a  priori,  as  follows.     Suppose  three  coordinates  x,  y,  and  z,  to  represent  the   partial  displacements 

of  any  molecule  M  in  their  respective  directions,  and  r  (=  v  *5  -f-  ys  -j-  z9)  the  total  displacement,  making  angles 
a,  ft,  7,  respectively  with  the  axes  of  the  x,  y,  z,  so  that  x  =  r  .  cos  a,  y  =  r  .  cos  ft,  2  =  r  .  cos  7.  Now,  since  in 
this  theory  we  assume  that  the  displacements  of  the  molecules  are  infinitely,  or  at  least  extremely  small  com- 
pared with  the  distances  of  the  molecules  inter  se,  it  is  evident  that  whatever  be  the  law  of  molecular  action,  the 
force  resulting  from  any  displacement  must  (ceeteris  paribus)  be  proportional  to  the  linear  magnitude  of  that  dis- 
placement, and  can,  therefore,  be  only  of  the  form  r  .  0,  where  0  is  some  unknown  function  of  the  angles  a,  ft,  7, 
Principle  of  or  their  cosines.  And,  moreover,  since  such  infinitely  small  displacements,  in  whatever  direction  made,  neither 
partial  dis-  alter  the  angular  position,  nor  distance  of  the  displaced  molecule  among  the  rest,  by  any  sensible  quantity,  all 
placements.  t(,e;r  forces  will  act  on  it  in  its  displaced  position  in  the  same  manner  as  before.  Hence  the  total  force  deve- 
loped by  the  simultaneous  displacements  x,  y,  z,  or  by  the  single  displacement  r  must  be  equivalent  to  (or  the 
statical  resultant  of)  the  three  which  would  be  developed  independently  by  the  several  partial  displacements 
x,  y,  z.  Now  the  force  originating  in  the  partial  displacement  x  alone  will  result  from  r0  by  making  r  =  x  and 
0  equa'  to  a,  where  a  is  the  same  function  of  1,  0,  0,  that  0  is  of  cos  c,  cos  ft,  cos  7.  a  therefore  is  a  con- 
stant depending  only  on  the  position  of  the  axes  of  the  x,  y,  z  with  respect  to  the  molecules  of  the  crystal. 
And  when  this  partial  force  =  a  x  is  resolved  into  the  directions  of  these  several  axes,  since  its  direction  (what- 
ever it  be)  is  determinate,  the  resolved  portions  can  only  be  of  the  form  A.X,  A!  x,  A."  x,  where  A,  A',  A"  are  in 
like  manner  dependent  only  on  the  position  of  the  coordinates  x,y,  z  with  respect  to  the  molecules,  and  not  at 
all  on  o,  ft,  7,  which  are  arbitrary,  and  where  A*  -j-  A'*  -}-  A."*  =  a5.  The  same  being  true  of  the  partial  forces 
brought  into  play  by  the  displacements  y  and  z,  it  follows  that  the  total  force  arising  from  the  displacement  r 
must  be  the  resultant  of  the  three  forces 


respectively  parallel  to  the  axes  of  the  x,  y,  z,  where  the  coefficients  are  independent  of  a,  ft,  7,  and  where,  in  like 
manner,  B*  +  B*  +  B''2  =  b1,  C8  -f-  C'a  -f-  C"1  =  c2.  But  we  have  x  =  r  .  cos  a,  y  =  r  .  cos  ft,  z  =  r  .  cos  7,  so 
that  if  we  put 

f  =  r    {  A  .  cos  a  -f-  B  .  cos  ft  +  C  .  cos  7  }  , 

f  —  r    {  A'  .  cos  a  -f  B'  .  cos  ft  +  C'  .  cos  7  }  , 
f  =  r    {  A",  cos  a  +  B".  cos  ft  +  C".  cos  7  }  , 

the  resultant  of//',/"  will  be  the  force  urging  the  displaced  molecule. 

999.  Now  these  forces  acting  in  the  directions   of  the  coordinates  may  each  be  decomposed  into  two,  one   in  the 

Expression    direction  of  the  displacement  r,  and  the  other  at  right  angles  to  it  in  the  planes  respectively  of  r  and  x,  r  and  y, 
of  the  elas-  r  an(J  z<  the  sum  of  the  former  will  be 
ticity  in  any  _,          ,  -.  .          .... 

assigned  F  =  /.  COS  a  -f  /'  .  COS  /3  +  /"  .  COS  7, 

'°n'      which  is  the  whole  force  tending  to  urge  the  displaced  molecule  directly  to  its  position  of  equilibrium.     The  latter 

ohl'S     1    t    wil1  be  respectively  equal  to/,  sin  «,/  .  sin  ft,  and/"  .  sin  7  ;  but  as  they  act,  although  in  one  plane,  yet  not  in 

the'dhrec-  °  tne  same  direction,  they  will  not  destroy  each  other,  unless  they  be  to  each  other  in  the  ratio  of  the  sines  of  the 

tion  of  dis-    angles  they  make  with  each  other's  direction.     But  it  is  evident,  that  since  a,  ft,  7  are  arbitrary,  this  condition 

placement,    cannot  hold  good  in  general,  because  it  furnishes  two  equations,  which,  taken  in  conjunction  with  the  relation 

cos  a»  _j-  cos  f?  +  cos  7"  =  1,  suffice   to  determine   a,  ft,  7.     Hence  it  follows,  that   the  displaced  molecule  is, 

except  in  certain  cases,  urged  by  the  elastic  forces  of  the  medium  obliquely  to  the  direction  of  its  displacement. 

1000  ^r>  Fresnel  next  goes  on  to  observe,  that  in  general  every  elastic  medium  has  three  rectangular  axes,  in  the 

Axes  of  '      direction  of  which,   if  a  molecule  be  displaced,   the  resultant  of  the  molecular  forces  urging  it  will  act  in  the 

elasticity      direction  of  its  displacement.     These  are  the  excepted  cases  just  alluded  to,  and  to  the  axes  possessing  this  pro- 

defined  and  perty     (which  he  regards   as   the    true   fundamental   axes   of  the    crystal,)    he  gives    the  name   of  Axes  of 

investigated  gkfc^ 

To  demonstrate  this  proposition  we  must  observe,  that,  by  mechanics,  in  order  that  the  resultant  of  three 
rectangular  forces/  /'./"shall  make  angles  a,  ft,  7  with  their  three  directions,  and  therefore  be  coincident  in  direc- 
tion with  r,  they  must  be  to  each  other  in  the  ratio  of  the  cosines  of  these  angles,  and  therefore  we  must  have  the 
following  equations  expressive  of  this  condition, 

/         cos  a      /         cos  a      /'   _  cos  ft 
'f  =  cosft  '   f~"  ~  cos  7  '   f"  =  cos  7' 

These  three  equations  are  in  general  equivalent  to  two  only,  but  when  combined  with  the  equation 
cos  Os  _|_  cos  ft'  _)_  cos  7*  =  1  resulting  from  the  geometrical  conditions  of  the  case,  they  suffice  to  determine 
a,  ft,  and  7  ;  and  if  we  put  u,  v,  w  for  the  cosines  of  these  angles,  furnish  the  following  system  of  equations 
which  every  axis  of  elasticity  must  satisfy. 

(Ait-f  Bv  +  Cw)v—-  (A'w+  B'u  +  C'  «.>)«• 
(Aw  +  Bc  +  Cw)w=(A"u  +  B"v  +  C"w)u 
(A'u  +  B'v+'.C'w)w  =  (\"u  4-  B"v  +  C"  w)  v 

M2  +  t!2  -f  W*  =   1. 


LIGHT.  541 

Light.          Suppose  by  elimination  we  have  derived  from  these  equations  the  position  of  one  axis  of  elasticity,  then  it  will     1'art  IV. 
— v— '  follow  of  necessity,  that  two  others  must  exist,  at  right  angles  to  it  and  to  each  other.     To  prove  this,  we  v— - v— - 
must  consider  the  connection  between  the  partial  forces  developed  by  any  displacement  of  the  molecule  M,  and 
the  molecular  attractions  and  repulsions  of  the  medium.    Let  0  be  the  action  of  any  molecule  d  m  on  M,  which  we  Three  exis 
suppose  to  be  exerted  in  the  direction  of  their  line  of  junction,  and  to  be  a  function  of  their  mutual  distance  p.  ^"yric^' 
Then,  if  we  suppose  M  displaced  by  any  arbitrary  quantities  S  x,  6  y,  &  z  (infinitely  small  in  comparison  with  />)  ang'lt,s  ,„ 
in  the  direction  of  the  three  coordinates,  we  have  each  other. 


d  <t>  x  y  z 

and  putting  0'  =  -~,     and  —  =  cos  X,     -  =  cos  /*,     —  =  cos  v, 

dp  p  p  p 

we  have  30  =  <j>'  .    ,  o  x  .  cos  X  -f-  3  y  .  cos  /t  +  S  p  .  cos  v  }  . 

Consequently,  since  the  force  of  the  molecule  d  m,  resolved  into  the  directions  of  the  coordinates,  is  respectively 
equal  to 

(0  +  S0)dm.—  ,    (0  +  50)  dm.  2-,     and(0+S0)dTO.—  , 
P  f  P 

the  sum  of  all  these  throughout  the  medium  will  be  the  total  action  on  M  ;  but  since  in  the  original  position  of 
the  molecule  M  it  is  in  equilibrio,  we  have 

.  —  —0,andf<pdm.  —  —  0, 
so  that  the  whole  action  of  the  medium  on  M  in  its  displaced  situation  will  be,  in  the  three  directions  General 

f—dm.d<j>,          f2-dm.$(t>,  f—dm.6<j>;  between  tne 

'    p  p  p  partial  elas- 

that  is,  in  the  direction  of  the  x, 

f  <j>'  dm  .  {  cos  X1  5  *  -f-  cos  /i1  .  5  y  +  cos  if  .  S  z  }  ; 

$x,  Sy,  &x,  are  the  partial  displacements  of  M  in  the  directions  of  the  coordinates,  and  are,  therefore,  the  same 
we  denoted  in  Art.  998  by  x,  y,  z.  Restoring  these  denominations,  we  see  that,  on  this  hypothesis,  (the  most 
natural  which  can  be  formed  respecting  the  mode  of  molecular  action)  the  coefficients  A,  B,  C,  can  be  no  other 
than  the  following, 

A  =y  0'  d  m  .  cos  Xs,     B  =  f  0'  d  m  .  cos  X  .  cos  /*,     C  =  f  0'  d  m  .  cos  X  .  cos  v  • 
and  by  similar  reasoning  we  find 

A!  =  J"  <f>'  d  m  .  cos  \  .  cos  /»,     B'  =  f  0'  d  m  .  cos  /t2,  C  '  =  f  <t>'  d  m  .  cos  /*  .  cos  v  ; 

A"  =:  f  0'  d  m  .  cos  X  .  cos  v,     B  '  =;  f  <j>'  d  m  .  cos  p  .  cos  v,      C''  =  f  0'  d  m  .  cos  va  ; 
and,  consequently,  the  following  relations  must  necessarily  subsist  between  these  coefficients 

B  =  A',     C  =  A",     C'  =  B". 

This  premised,  suppose  we  have  determined  one  axis  of  elasticity  of  the  medium  by  the  foregoing  equations.      1002. 
Since  the  positions  of  the  axes  of  the  coordinates  are  arbitrary,  we  are  at  liberty  to  suppose  that  of  the  x  coin- 
cident with   the   axis  so  determined,  which   renders  A'  =  A"  ~  0,  and  consequently  B  =  0   and   C  =  0,   and 
Bv  =  C',  because  the  relations  above  demonstrated  are  general  and  independent  of  any  particular  situation  of 
the  axes.     The  equations  of  Art.  1000  then  become  One  axis 

A  u  v  =  (B'  v  +  C'  w)  u,     PLUW=  (B"  v  +  C"  w)  v,  JfiSStoi 

(B'l)  +  C'w)w  —  (C'u-f  C"W)V,      W8-f  »«  +  «#=:  I.  of  the  other 

determined, 

Now  if  we  put  M  =  0,  or  a  =  90°,  the  two  former  of  these  are  satisfied  without  any  relation  supposed  between 
r  and  w,  so  that  if  we  determine  these  from  the  two  latter  only,  the  whole  system  will  be  satisfied.  These 
(making  u  =  0)  give  at  once  by  elimination 


where  m  =  ^  -  —  (.     Now  since  »i8  is  necessarily  positive,  4  ms  -f-  1  is  so,  and  is  >  1  ;  therefore  — 

v  4  wi2  -j-  1 

is  real  and  <  1,  consequently  w1  and  t>8  are  both  positive,  and  therefore  v  and  w  both  real,  and  less  than  unity. 
Hence  it  follows,  that  there  are  necessarily  two  axes  at  right  angles  to  the  x  which  satisfy  the  conditions  of  axes 
of  elasticity,  and  the  opposite  signs  of  t>  and  w  show  that  they  are  at  right  angles  to  each  other. 

For  simplicity,  therefore,  we  will  in  future  suppose  the  directions  of  the  coordinates  to  be  coincident  with  those 
of  the  axes  of  elasticity,  so  as  to  make 


542  LIGHT. 

A  =  0,  A'  =  A"  =  0;     B'  =  b,  B  =  B"  =  0  ;     C"  =  c,  C  =  C'  =  0  ;  Part  IV. 

then  we  have  by  Art.  998  for  the  partial  forces,  v— - v— - • 

f  =  a  x  =  ar .  cos  a,      f  =  6  y  =  h  r  .  cos  ft,      f"  =  c  z=  cr  .  cos  <y, 
and  by  999, 

F  =  r  {a.  cos  a?  +  b  .  cos  /3s  +  c .  cos  -/'  } 

for  the  whole  force  urging  the  molecule  M  in  the  direction  of  the  r,  generally  assumed,  in  which  it  will  be 
observed  that 

a  =f<j>' .  cosX'rfm,       b  —  J"  <j>' .  cos  p.*  d  m,       c  =  J"  <p' .  cos  v*  d  m. 

1004.  M.  Fresnel  next  conceives  a  surface,  which  he  terms  the  "  Surface  of  Elasticity,"  constructed  according  to  the 
The  surface  following  law: — on  each  of  the  axes  of  elasticity,  and  on  every  radius  r  drawn  in  all  directions,  take  a  length 

ofe/aitidty  proportional  to  the  square  root  of  the  elasticity  exerted  on  the  displaced  molecule  by  the  medium  in  the  direc- 

denned  and  '  . 

investigated  tion  of  the  radius,  or  to  v  F.     Then  if  we  call  R  this  length,  or  the  radius  vector  of  the  surface  of  elasticity,  we 
shall  have 

11'  =  {  a  r  .  cos  u-  -f-  b  r  .  cos  /3*  -{-  c  r .  cos  7*  }  X  const. 

Its  radius     The  values  of  R  parallel  to  the  axes  are  then  had  by  the  equation 
vector  ex- 
pressed. R8  =  const  ar,     R"  =  const  X  b  r,     R*  =  const  x  cr 

which  (for  brevity,  as  we  shall  have  no  further  occasion  to  recur  to  our  former  denominations)  we  shall  express 
simply  by  a',  b*,  c8,  so  that  the  equation  of  the  surface  of  elasticity  will  be  of  the  form 

R!  =  a1 .  cos  X'  +  6* .  cos  Y'  -f  cs .  cos  Z«, 

where  X,  Y,  Z,  now  stand  for  a,  /3,  7,  the  angles  made  by  R  with  the  axes  of  the  coordinates. 

1005.  Let  us  now  imagine  a  molecule  displaced   and  allowed  to  vibrate  in  the  direction   of  the  radius    R,  and 
Velocity       retained  in  that  line,  or  at  least  let  us  neglect  all  that  part  of  its  motion  which  takes  place  at  right  angles  to 
o"  olariza-  tne  ra(^'us  vector.     Then  the  force  of  elasticity  by  which  its  vibrations  are  governed  will  be  proportional  to  Rs, 
tion  of  an     and  the  velocity  of  the  luminous  wave  propagated  by  means  of  them,  in  a  direction  transverse  to  them  (or  at 
interior        right  angles  to  R)  will  be  proportional  to  R,  so  that  the  surface  of  elasticity  being  known,  the  velocity  of  a  wave 
wave  deter-  transmitted  through  the  medium  in  a  given  direction,  and  with  a  given  plane  of  polarization  will  be  had  at  once 

as  follows.  Parallel  to  the  surface  of  the  wave,  and  at  right  angles  to  its  plane  of  polarization  draw  a  straight 
line.  This  will  be  the  direction  of  the  vibrations  by  which  the  wave  is  propagated.  Parallel  to  this  line  draw  a 
radius  vector  to  the  surface  of  elasticity,  and  it  will  represent  the  wave's  velocity. 

1006.  The  equation  of  the  surface  of  elasticity,  if  we  put  for  R,  cos  X,  cos  Y,  cos  Z,  their  values  in  terms  of  three 
Equation  of  coordinates  will  become 

the  surface  rxt  -f-  «*  4-  2")'  =  a'  a8  jf  6*  Vs  -f  C*  Z\ 

of  elasticity. 

It  is,  therefore,  in  general  a  surface  of  the  fourth  order.  If  we  suppose  it  cut  by  a  plane  passing  through  its 
centre,  whose  equation  must  therefore  in  general  be  of  the  form  mx-\-  ny  +  pz  =  0,  the  curve  of  intersection 
will  be  a  species  of  oval  whose  diameters  are  not  necessarily  all  equal. 

1007.  Suppose  now  any  molecule  set  in  vibration  in  this  plane,  then  at  any  period  of  its  motion  it  will  not  be  urged 
Resolution    directly  to  its  point  of  rest  but  obliquely,  so  that  it  will  not  describe  a  straight  line,  but  will  circulate  in  a  curve 
of  an  inci-    more  or  less  complicated ;  its  motion  in  this,  however,  will  always  be  resolvable  into  two  vibratory  rectilinear  ones  at 
dent  wave    r;ght  angles  to  each  other,  one  parallel  to  the  greatest,  and  the  other  to  the  least  diameter  of  the  section.    Each  of 

W°'      these  vibratory  motions  will,  by  the  laws  of  motion'be  performed  independently  of  the  other,  and  therefore  the  motion 

propagated  through  the  crystal  will  aftect  every  molecule  of  it  in  the  same  way  as  if  two  separate  and  independent 

Polari:ed  in  rectilinear  vibrations  (at  right  angles  as  above)  were  propagated  through  it,  with  different  velocities.  Consequently 

opposite       every  system  of  waves  propagated  from  without  into  the  crystal,  will  necessarily  on  entering  it  be  resolved  into  two 

planes.         propagated  w4th  different  velocities,  and  polarized  in  planes  at  right  angles  to  each  other,  viz.  those  parallel 

respectively  to  the  greatest  and  least  diameter  of  a  section  of  the  surface  of  elasticity  parallel  to   the  plane  of 

either  wave.     And  as  every  difference  in  the  velocities  of  two  waves  propagated  parallel  to  each  other  thro-Jgh 

a  medium,  gives  rise  to  a  corresponding  difference  in  their  planes  at  their  emergence  from  it  into  another,  where 

they  assume  a  common  velocity,  these  waves  will  at  their  egress  no  longer  be  parallel,  and  the  rays  which  arc 

perpendicular  to  them  will  be  inclined  to  each  other,  thus  producing  the  phenomena  of  double  refraction ;  and  it 

is  evident  that  the  waves  at  their   egress  must  retain  the  planes  of  polarization  they  received   in  the  crystal, 

because  any  molecule  of  the  exterior  medium  at  the  junction  of  the  media  will  begin  to  move  only  in  the  plane 

in  which  it  was  displaced  by  the  contiguous  molecule  in  the  medium. 

1008.  This  theory  then  accounts  perfectly  both  for  tire  bifurcation  of  the  emergent  ray,  and  the   opposite  polariza- 
tions of  the  two  portions  into  which  it  is  divided.     These  portions  will  coincide  in  direction,  and  there  will   be 
no  double  refraction  when  the  section  of  the  surface  of  elasticity  above  mentioned  is  (if  such  can  ever  be  the 
case)  a  circle,  because  all  its  radii  being  then  equal,  the  elasticity  is  the  same  in  all  directions,  and  all  vibrations 
performed  in  it  will  have  equal  periods,  so  that   in  this  case  the  resolution  of  the  incident  wave  into  two  no 
longer  takes  place,  nor  is  its  plane  of  polarization  changed.     Now  the  section  in  question  becpmes  a  circ  e, 
when  *»  +  y8  +  2"  =  const  =  r»,  or  when  a'  X*  -(-  6!  y'  +  ca  z«  =  r4.  Combining  these  with  m  x  +  n  y  +  p z  =  0, 
we  get 


LIGHT.  543 

Light.  r<  =  rz  (o?J  +  y1  +  z*),  p»rt  3V- 

p»  r4  =  r*  (/  **  +  p*  y8  +  (m  *  +  «  ?)*), 
and  J92  r*  =:  p*  a'x*  +  p*  b*y* +  <?  (mx  +  n  y}', 

and  equating  these,  and  considering  that  the  equation  thence  resulting  ought  to  be  verified  independently  of  any 
particular  values  of  x,  y,  we  get 

r2  (w2  +  p')  =  aV  +  ms  c2,  Investiga- 

tion of  the 
m  71  r4  =  m  n  c!,  optic  axes. 

,.z  (p>  +  n^  —  bip>  +  n2  c2. 
These  equations  cannot  be  satisfied  except  by  supposing  either  m,  n,  or  p  to  vanish,  or  the  section  in  question  to 

pass  through  one  or  other  of  the  axes.    If  we  suppose  m  =  0,  we  have  r  —  a,  (  —  J  =  — — — t,  which  shows  that 

(n\t  n 

—  I  cannot  be  positive,  and  of  course  —  not  real,  unless  a,  the  semiaxis  of  the  surface  through  which  the 
P/  P 

section  passes,  be  that  intermediate  in  length  between  6  and  c,  the  other  two  semiaxes. 

It  appears  then,  that  the  surface  of  elasticity  admits  of  two  circular  sections  and  no  more,  formed  by  diametral      1009 

7i 

planes  passing  through  the  mean  axis  of  the  surface,  and  (since  —  has  two  values  equal  but  of  opposite  signs) 

that  these  sections  are  both  equally  inclined  to  each  of  the  other  two  axes.  The  normals  to  these  sections  are 
the  directions  of  no  double  refraction,  or  the  optic  axes  of  the  crystal.  Of  these,  then,  there  will  be  two  and  two 
only,  in  all  crystals  which  possess  three  unequal  axes  of  elasticity,  and  rays  propagated  along  them  will  suffer 
neither  double  refraction,  nor  change  of  polarization. 

The  position  of  these  axes  depends  wholly  on  the  values  of  a,  b,  c,  the  semiaxes  of  the   surface  of  elasticity.      1010. 
We  have,  however,  no  other  measure  of  the  elasticity  of  the  medium  than  the  velocity  with  which  the  rays  are  Dispersion 
propagated  through  it ;   and  if,  as  the  phenomena  of  ordinary  dispersion  indicate,  the  rays  of  different  colours  be  °f 'he  axes 
propagated  in  one  and  the  same  medium  with  velocities  somewhat  different,  (an  effect  which  might  result  from  °ojj^rent 
certain  suppositions  relative  to  the  extent  of  the  sphere  of  action  of  its  molecules  compared  with  the  lengths  of  explained, 
an  undulation,)  the  semiaxes  a,  b,  c,  which  must  be  taken  proportional  to  the  velocities  of  propagation,  must  be 
supposed  to  vary  a  little  for  waves  of  different  lengths.     Now  this  variation  may  not  be  in  the  same  ratio  for  all 

It  71 

the  three  semiaxes,  and  thus  a  variation  in  the  values  of  —  will  arise.     But  —  is  the  tangent  of  the  inclination 

P  P 

of  the  plane  of  section  to  the  plane  of  the  x  y,  or  of  half  the  angle  the  two  circular  sections  make  with 
each  other,  t.  e.  the  cotangent  of  half  the  angle  between  the  optic  axes,  which  will  thus  vary,  and  give 
rise  to  that  separation  of  axes  of  different  colours,  and  their  distribution  over  a  certain  angle,  in  the  plane 
containing  any  two  of  the  same  colour,  which  observation  shows  to  exist,  (Art.  921  and  922.) 

The  general  laws  of  double  refraction  flow  with  great  facility  from  these  principles.      We  have  only  to      JQJJ 
resume  the  construction  and  reasoning  of  Art.  806  and  807,  et  seq.,  substituting  for  the  ellipsoid  of  revolution,  Application 
which  the  Huygenian  theory  assumes  as  the  figure  of  a  wave  originating  in  any  molecule  of  the  crystal,  the  of  the  Hu\ 
surface,  whatever  it  be,  which,  in  the  general  case,  terminates  a  wave  so  propagated,  and  investigating  the  point  genian  con- 
of  contact  I  (fig.  170)  of  this  surface  with  a  plane  IKT  passing  through  the  line  KT  drawn  as  there  described.  s'ruct!< 
There  is  this  difference,  however,  in  the  two  cases,  or,  at  least,  in  the  method  of  treating  them,  that  in  the  case86" 
theory  there  stated  the  form  of  the  wave  is  made  a  matter  of  arbitrary  assumption,  in  the  present  case  it  is 
to  be  determined  a  priori.     This  will  render  it  necessary  to  depart  in  some  respects  from  the  course  before 
adopted.     If  we  know,  a  priori,  the  form  of  the  wave,  the  position  of  the  tangent  plane  is  given  ;  vice  versd, 
if  we  can  determine  the  position  of  this  plane  in  all  cases,  a  priori,  the  figure  of  the  wave,  which  must  be 
such  as  to  touch  all  such  planes,  under  the  conditions  of  the  case,  becomes  known. 

Now,  in  Art.  807,  it  is  shown  that  the  tangent  plane  is  in  all  cases  coincident  with  the  position  assumed     1012 
within  the  crystal,  by  the  surface  of  a  plane  indefinite  wave  propagated  from  an  infinitely  distant  luminary,  per-  Direction 
pendicular  to  the  line  of  incidence  R  C.     It  follows,  moreover,  from  Art.  81 1,  that  if  we  know  the  velocity  with  and  velocity 
which  such  a  plane  wave  advances  within  the  crystal  in  a    direction    perpendicular  to  its    surface,  we  mayofaPlan* 
calculate- its  inclination  to  the  surface  of  incidence  by  the  law  of  ordinary  refraction,  assuming  an  index  of"a*c" 
refraction  which  is  to  that  of  the  ambient  medium  as  the  velocity  of  the  wave  before  incidence  is  to  its  velocity 
within  the  medium  perpendicular  to  its  own  surface.     The  reader  will  here  keep  in  view  the  distinction  noticed 
in  Art.  813  between  the  velocity  of  the  wave  and  that  of  the  ray  conveyed  by  it,  whose  direction,  generally 
speaking,  is  oblique  to  its  surface.     Now  the  velocity  of  a  wave  within  the  medium  in  any  direction  is  given 
by  the  equation  of  the  surface  of  elasticity,  whose   radius  vector   expresses   it  in  all  cases.     But  it  has  been 
shown,  that  every  vibration  impressed  on  the  molecules  of  the  crystal  is  resolved  into  two  rectilinear  ones  propa- 
gated with  velocities  proportional  to  the  greatest  and  least  diameters  of  that  section  of  the  surface  of  elasticity 
which  is  parallel  to  the  plane  in  which  they  are  performed.    Now  it  is  the  same  thing,  (as  far  as  the  law  of  double 
refraction  is  concerned,)  whether  we  regard  the  bifurcation  to  take  place  by  the  separation  of  a  single  exterior 
ray  into  two  interior  ones,  or  a  single  interior  into  two  exterior.     We  will  take  the  latter  case,  and  suppose  the 


544  LIGHT 

Light.      ordinary  and  extraordinary  plane  waves  to  be    parallel  within  the    medium.      Their  velocities  may  then  be 
v— - ^s~ms  investigated  as  follows  :  the  equation  of  the  surface  of  elasticity  being 
Velocities 

of  an  ordi-  R4  =  a"-  X*  +  b*  yj  +  c8  Z1, 

nary  and 

extraordi-     if  we  take,  for  the  equation  of  the  second  plane, 
nary  plane 
wave  inves-  z  =  m  x  +  n  y, 

and  put  V  for  the  maximum  or  minimum  radius  vector  of  the  surface  in  the  section  in  question,  V  will  be  the 
value  of  11,  which  makes  d  R  =  0,  and  therefore  will  be  given  by  elimination  from  the  following  system  of 
equations 

V"  =  **  +  y*  +  **, 

V«  =  of  x-  +  6*  j/2  +  c*  z\ 

z   =  m  x  -f-  n  y, 

and  their  differentials,  regarding  V  as  constant.  This  elimination,  which  is  complicated  enough,  must  be  con- 
ducted as  follows  :  first,  if  among  the  differential  equations  we  eliminate  dx,  dy,  dz  ;  and  for  z  in  the  whole 
system  substitute  its  value,  we  shall  get,  putting p  =  a*  —  6s;  q  =  a"  —  c*;  r  =  bl  —  c8; 

V«  =  (a8  +  m*  c1)  X*  +  (6s  +  n'  (?)  y*  +  2  m  n  c8  x  y, 
V  =  (1  +  ?ns)  *"  +  (1  +  «*)  y*  +  Zmnxy, 
0  =  mnq.i?  -  mnr  y'  +  k  x  y, 
where  k  =  p  +  n'  q  -•    m*  r  —  (1  -f-  if)  q  —  (1  -f-  m')  r. 

These,  by  elimination,  give  the  following,  in  which 

M  =  A»  +  4  ms  ne  q  r ; 

M  x1  =  Vs  (V  —  c')  {  (I  +  71")  A  +  2  m1  n*  r  }  -r  k  V, 
M  y*  =  -  Vs  (V  -  c*)  {  (1  -f  m«)  k  -  2  m*  w-  q}  +  rq  V*, 
Mcry=  -mre{  (1  +  n")  9+  (1  +  >«•')  r  }  V«(V*  -  c*)  -f-  2wi7zgrV«; 

and  by  equating  the  square  of  the  last  of  these  to  the  product  of  the  two  first,  we  find,  after  all  reductions,  the 
following  equation  for  determining  V  : 

(V2  -  «2)  (V»  -  b1)  +  TO*  (V*  —  62)  (V»  -  c8)  +  wa  (V«  -  a8)  (Vs  -  c!)  =  0. 

1013.  The  roots  of  this  equation  determine  the  maximum  and  minimum  values  of  the  radius  vector  in  the  plane  of 
(jeireral       section,  and  therefore  the  velocities  of  ordinary  and  extraordinary  plane  waves  moving  parallel  to  each  other 
eqmtion  of  w;tnjn   tne  crystal,  and  these  found,  the  figure  of  the  wave  becomes  known,  from  the  condition  that  its  surface 
mMteJ     "  must  always  be    a  tangent  to   a  plane  distant  by  the  quantity  V  from  the  secant  plane  whose  equation  is 
lio!n  a         ~— mx-\-ny;    and  that,  whatever  be  the  values  of  m,  and  n.     Its  investigation  is  therefore  reduced  to  a 
point  in  the  purely  geometrical  problem.     Required  the  equation  of  a  curve  surface,  which  shall  touch  every  plane  parallel 
medium.       (O  a  plane  whose  equation  is  z  —  m  x  -\-  ny  ;    and  distant  from  it  by  a  quantity  V,   a  function  of  m  and  n 

given  by  the  above  equation,  which,  being  resolved,  will  be  found  to  lead  to  the  following  equation 

(a2  jr8  +  ft'i/*  +  <?  z«)  (**  +  y1 -\-  z5)  -  as  (i!  +  c8)  x1  -  62  (a8  +  c!)  y*\  _ 
-  c8  («8  +  K)  z2  +  a2  6!  c8  j  ~ 

1014.  The   surface  represented  by  this  equation  is,   generally  speaking,  of  the  fourth  order,  and  consists  of  two 
Nonexist-  distinct  surfaces,  or  sheets,  (nappes.)     One  of  these,  by  its  contact  with  the  plane  in   question,  determines  the 
enre  of  the  direction  of  the  ordinary,  and  the  other  of  the  extraordinary  ray.     Now,  it  is   important  to   remark,   that  this 
UtTof're-  ecluat'on>  so  'onS  as  particular  values  are  not  assigned  to  a,  b,  c,  is  not  decomposable  into  quadratic  factors,  so 
fraction  in  that  neither  of  the  sheets  of  which  it  consists  is  spherical,  or  ellipsoidal  ;  and,  consequently,  neither  the  ordinary 
liiaxal  nor  the  extraordinary  ray  follows  either  the  Cartesian  or  Huygeiiiaii  law  of  refraction.     This  is  a  consequence 
-rystals.  too  remarkable  not  to  have  been  put  to  the  test  of  experiment.     Two  methods   have  been  put  in  practice  by 

M.  Fresnel  for  this  purpose.  The  first  consisted  in  measuring  directly  the  velocities  of  the  two  rays  in  plates  of 
topaz  cut  in  different  directions  with  respect  to  their  axes  by  the  method  explained  under  the  head  of  inter- 
ferences, (Art.  738  and  739.)  Since  a  difference  of  velocity  of  the  interfering  rays  displaces  the  diffracted  fringes 
as  a  difference  of  thickness  would  do,  it  is  manifest  that  if,  in  two  plates  differently  cut,  but  of  precisely  the 
tame  thickness  the  fringes  formed  by  the  ordinary  rays  are  differently  displaced  when  the  plates  are  combined 
successively  with  one  and  the  same  equivalent  plate  of  glass,  or  any  other  standard  medium,  their  velocity  cannot 
be  the  same  in  both  plates;  and  if  such  difference  be  observed  to  take  place,  both  in  the  fringes  formed  by  the 
interference  of  the  ordinary  and  of  the  extraordinary  rays  severally,  with  a  compensated  pencil,  it  is  clear  that 
neither  can  have  a  constant  velocity.  Now  the  condition  of  equal  thickness  is  secured  by  cementing  the 
two  plates  edge  to  edge,  and  grinding  and  polishing  them  together,  and  carefully  examining  the  surfaces  after 
the  operation,  to  be  satisfied  of  their  precise  continuity,  which  may  be  done  by  the  reflected  image  of  a  distant 
object,  and  yet  more  delicately  by  pressing  slightly  on  them  a  convex  lens  of  long  focus,  over  their  line  of 
junction.  If  the  coloured  rings  formed  between  the  surfaces  be  uninterrupted,  we  are  sure  that  this  condition 


L  I  G  II  T  5  15 

is  rigorously  satisfied.     The  experiment  so  made,   M.  Fresnel  found  to  confirm  the  conclusion  to  which  the     ^n  IV. 
above  theory  leads.     But  in  corroboration  of  this  important  result,  the  following  method  was  also  used.  v—  "v—  *' 

In  topaz  the    extraordinary  refraction  is  stronger  than  the  ordinary  ;    so  that  the  ordinary  ray,  when  the      1 
two  are  separated   by  a  prism    of   that  medium,   may  be  at  once    recognised,    by  being    the  least  deviated.  p"°^tet* 
M.  Fresnel  procured  two   prisms  to  be  cut  from  one   topaz,  in  both  of  which  the  base  was  parallel  to  the  j,rove  lhe 
cleavage  planes,  and  therefore  perpendicular  to  a  line  bisecting  the  angle  between  the  optic  axes  and  to  the  same. 
principal  section  of  the  crystal,  i.  e.  to  the  mean  axis  of  elasticity  ;  but  in  one  the  plane  of  the   refracting 
angle  was  coincident  with,  and  in  the  other  perpendicular  to,  that  section,  these  being  the  planes  in  which  the 
difference  between  the  velocities  of  the  ordinary  ray  is  the  greatest,  as  is  easily  seen  from  what  has  above  been 
said.     These  prisms  were  cemented  side  by  side,  so  as  to  have  their  bases  in  one  plane  and  their  refracting  edges 
in    one    straight    line  ;    and  were    then   very  carefully  ground   and    polished    to    plane    surfaces,  so  t'hat  the 
refracting  angles  in  both  could  not  be  otherwise  than  precisely  equal.     In  this  situation  the  compound  prism 
ABC,  fig.  199,   1,  (which   is  seen  in  perspective  in  fig.  199,  2,)  whose  refracting  angle  ABC  was  about  92°, 
was  achromatized  by  two  prisms  C  B  A  and  D  C  A  of  crown  glass,  in  which  circumstances  a   slight,  uncompen- 
sated  refractionrremained  in  favour  of  the  topaz  prism.     Looking  now  through  the  side  E  B,  the  whole  comb- 
nation  was  turned  round  the  refracting  edge  as  an  axis,  till  the  image  of  a  distant  object,  a  black  line  on  a 
white  ground,  appeared  stationary  ;  so  that  the  refracted  rays,  both  ordinary  and  extraordinary,  must  have  tra- 
versed the  prisms  very  nearly  parallel  to  the  base,  or  at  right  angles  to  the  mean  axis,  but  in  the  different  planes 
above  mentioned  in  each.     Now  it  was  observed,  that  the  least  refracted  image  of  the  black  line  so  seen,  that 
is  the  ordinary  one,  was  broken   at   the  junction  of  the  two  prisms,  being  more  deviated  by  one  than  by  the 
other,  while  the  most  refracted  or  extraordinary  image  formed   a  continuous  line  in   both.     This  latter  fact 
(whith,  at  first  sight,  would  lead  us  to  suspect  that  the  extraordinary  image  had  been  mistaken  for  the  ordinary 
one)  is  a  consequence  of  the  theory  above  explained,  and  is  an  additional  confirmation  of  it. 

When  two  of  the  axes  of  elasticity  (as  6  and  e,  for  instance)  are  equal,  the  general  equation  of  the  surface  of      1016. 
the  wave  becomes  decomposable  into  two  factors,  and  may  be  put  under  the  form  Case  of 


which  is  the  product  of  the  equation  of  a  sphere  with  that  of  an  ellipsoid  of  revolution.  In  this  case  the  two 
circular  sections  coincide  with  the  plane  of  the  y  z,  and  the  two  optic  axes  with  the  axis  of  the  x.  We  have 
here  then  the  case  of  \iniaxal  crystals,  and  are  thus  furnished  with  an  A  priori  demonstration,  both  of  the  Huy- 
genian  law  of  elliptic  undulations,  in  the  case  of  the  extraordinary  wave  in  such  crystals,  and  of  the  constancy 
of  the  index  of  refraction  in  that  of  the  ordinary.  The  manner  in  which  this  results  as  a  corollary  from  the 
general  case  is  at  once  elegant  and  satisfactory. 

M.  Fresnel  gives  the  following  simple  construction  for  the  curve  surface  bounding  the  wave  in  the  case  of     1017. 
unequal  axes,  which  establishes  an  immediate  relation   between  the  length  and  direction  of  its  radii.     Conceive  Constnic- 
an  ellipsoid  having  the  same  semiaxes  a,  b,  c  ;  and  having  cut  it  by  any  diametral  plane,  draw  perpendicular  ^°"v°  ^  ^.t 
to  this  plane  from  the  centre  two  lines,  one  equal  to  the  greatest,  and  the  other  to  the  least,  radius  vector  of  the  ellipsoid. 
section.     The  loci  of  the  extremities  of  these  perpendiculars  will  be  the  surfaces  of  the  ordinary  and  extraordi- 
nary waves  ;  or,  in  other  words,  their  lengths  will  be  t'he  lengths  of  the  radii  of  the  waves  in  those  directions, 
and  will  therefore  measure  the  velocity  of  the  two  rays  propagated  in  those  directions,  in  the  same  way  as  the 
radii  of  the  Huygenian  ellipsoid  are  proportional  to  the  velocities  of  the  extraordinary  ray  in  their  direction. 

Finally,  if  we  divide  unity  by  the  squares  of  the  two  semiaxes  of  a  diametral  section  of  the  ellipsoid,  the      1018. 
difference  of  these  quotients  will  be  found  to  be  proportional  to  the  product  of  the  sines  of  the  angles  which  Origin  of 
the  perpendicular  to  this  section  makes  with  the  two  normals  to  the  planes  of  the  circular  sections  of  the  ':"    "   ,° 
ellipsoid.     Now,  in  all  the  crystals  hitherto  known,  these  sections  differ  very  little  from  the  circular  sections  of  of  ^"two 
the  surface  of  elasticity,  and  may,  without  sensible  error,  be  supposed  to  coincide  with  them  ;  consequently,  the  sines. 
two  normals  in  question  may  be  taken  for  this  purpose  as  the  optic  axes  of  the  crystal.     We  have  thus  the 
origin  of  that  law,  deduced  from  the  phenomena  of  the  coloured  lemniscates,   which  makes  the  difference   of 
the  squares  of  the  reciprocal  velocities  proportional  lo  the  product  of  the  sines  made  by  the  ray  with  the  optic 
axes  ;  and  thus  the  phenomena  of  the  polarized  rings  are  all  made  to  depend  on  the  same  general  principles. 

Such  is  the  beautiful  theory  of  Fresnel  and  Young,  (for  we  must  not  in  our  regard  for  one  great  name  forget  1019 
the  justice  due  to  the  other,  and  to  separate  them  and  assign  to  each  his^hare  would  be  as  impracticable  as  invi- 
dious, so  intimately  are  they  blended  throughout  every  part  of  the  system  ;  early,  acute,  and  pregnant  suggestion 
characterising  the  one,  —  and  maturity  of  thought,  fulness  of  systematic  developement,  and  decisive  experimental 
illustration,  equally  distinguishing  the  other.  If  the  deduction  in  succession  of  phenomena  of  the  greatest  variety 
and  complication  from  a  distinctly  stated  hypothesis,  by  strict  geometrical  reasoning,  through  a  series  of  inter- 
mediate steps,  in  which  the  powers  of  analysis  alone  are  relied  on,  and  whose  length  and  complexity  is  such 
as  to  prevent  all  possibility  of  foreseeing  the  conclusions  from  the  premises,  be  a  characteristic  of  the  truth 
of  the  hypothesis,  —  it  cannot  be  denied  that  it  possesses  that  character  in  no  ordinary  degree  ;  but,  however 
that  may  be,  as  a  generalization  the  reader  will  now  be  enabled  to  judge  whether  the  encomium  we  passed  on 
it  in  a  former  Article  be  merited.  We  can  only  regret  that  the  necessary  limits  of  this  Essay,  which  is  already 
extended  greatly  beyond  our  original  design,  forbid  our  entering  farther  into  its  details. 

The  axes  of  elasticity  are  those  which  M.  Fresnel   regards   as  the  fundamental   axes  of  a  doubly  refractive      1020. 
medium.     The  optic  axes  can  in  no  view  of  the  subject  be  regarded  as  such,  for  several  obvious  reasons.     First,  Dr.^Brew- 
they  are  seldom  symmetrically  situated  relative  to  fundamental  lines  in  the  crystalline  form  ;  sec  ndly,  because  stfer  s  '"^"f 
they  vary  in  position  according  to  the  colour  of  the  incident  light  ;  thirdly,  because  it  is  found  that  for  one  and  z;n|*  U£C!>< 
the  same  coloured  illumination,  and  in  the  same  crystal,  their  situation  varies  by  a  variation  of  temperature. 

VOL.  iv  4  B 


546  LIGHT. 

Light.  This  important  fact  has  been  lately  ascertained  by  M.  Mitscherlich,  and  we  shall  presently  have  occasion  to  speak  Part  IV. 
v^-^y^— '  further  of  it.  Prom  all  these  reasons  it  follows,  that  we  can  regard  them  only  as  resultant  lines,  to  which  no  v»v— 
&  priori  properties  can  be  supposed  to  belong,  but  which  simply  satisfy  the  condition  v  —  if  =  0,  according  to 
the  laws  which  regelate  the  constitutions  of  the  functions  v,  tf,  the  velocities  of  the  two  rays,  in  terms  of  those 
quantities  which  we  may  regard  as  fundamental  data,  and  the  situation  of  the  ray  within  the  medium.  The  axes 
of  elasticity  themselves  may,  perhaps,  be  regarded  as  mere  resultants  from  the  equations  of  Art.  1000,  and 
determined  from  other  remoter  data  dependent  on  the  fundamental  lines  in  the  crystalline  form,  and  the  intensity 
and  distribution  of  the  molecular  forces  within  it.  Accordingly,  Dr.  Brewster  considers  the  optic  axes  as  the 
resultants  of  others  which  he  terms  polarizing  axes,  and  from  which  he  conceives  to  emanate  polarizing  forces 
producing  the  phenomena  of  the  rings  and  of  the  double  refraction  and  polarization  observed.  We  shall  not 
here  stop  to  examine  into  the  propriety  of  these  terms.  The  reader  who  may  have  doubts  on  the  subject  will, 
in  what  follows,  mentally  substitute  other  and  more  general  phrases  in  their  place  expressive  of  relation  and 
causality,  while  we  proceed  to  state  the  assumptions  with  which  he  sets  out,  and  the  conclusions  he  very  inge- 
niously deduces  from  them. 

1021.          Postulate  1.    A  polarizing  axis,  when  single,  has  the  characters  of  an  axis  of  no  double  refraction,  and  is 
A  single       coincident  with  the  axis  of  the  Huygenian  spheroid  in  such  crystals  as  have  but  one.     A  positive  axis  acts 
polar::ing     as  the  axis  in  quartz,  &c.  may  be  supposed  to  do,  and  a  negative,  as  that  of  carbonate  of  lime,  &c. 
"  Post.  2.  The  polarizing  force  of  a  single  axis  in  any  medium  is  proportional   to,  and  measured  by,  the  tint 

developed  in  the  ordinary  and  extraordinary  pencils  into  which  a  doubly  refracting  prism  analyzes  a  polarized 
.  rav-  which  has  traversed  a  given  thickness  of  the  medium. 

1023.  Carol.  1.  The  polarizing  force  of  a  single  axis  in  the  same  medium  is  as  the  square  of  the  sine  of  the  angle 
made  by  the  ray  traversing  it  internally,  with  the  axis. 

1024.  Carol.  2.  The  same  force  is  also  inversely  as  the  thickness  necessary  to  be  traversed  at  a  given  angle  to 
develope  the  same  or  equal  tints.     This  may  be  regarded  as  the  intrinsic  polarizing  force  or  intensity  of  the  axis. 

1025.  Post.  3.  When  two  axes  exist  in  one  medium  and  operate  together,  they  polarize  a  tint  whose  measure   (see 
Composi-      Art.  906)  is  the  diagonal  of  a  parallelogram  whose  sides  measure,  on  the  same   scale,  the  tints  which  would  be 
tion  of  tints  polarized  by  either,  separately,  and  include  between  them  an  angle  double  of  the  mutual  inclination  of  two  planes 
in  the  case    pass;ng  through  the  ray  and  either  axis  respectively. 

°  'lOae"*       Carol.  1.  If  t  and  t'  be  the  numerical  measures  of  the  tints  polarized  by  either  of  two  axes  separately,  T  that 
Formula  for  P°'ar'zed  by  their  joint  action,  and  C  the  angle  between  the  planes  just  described,  the  tint  T  will  be  given  by  the 

the  com-      equation  T*  =  2J  -f  2  tt' .  cos  2  C  +  if*. 

pound  tint. 

1027  Carol.  2.  If  a  and  b  represent  the  intensities  of  the  axes,  and  a  and  /3  the  angles  which  the  ray  makes  with 

each  respectively,  we  have  t  =  a  .  sin  a2 ;  t'  =  b  .  sin  /J*,  and 

Ta  =  0  .  sin  a')»  +  (6  .  sin  /3s)'  +  2  a  6  .  sin  o* .  sin  /3* .  (1  -  2 .  sin  C«), 

=  {  a  .  sin  a*  +  6  .  sin  j3*  Js  —  4  a  b  .  sin  a* .  sin  /3s .  sin  C4, 
or  else  T*  =  {  a  .  sin  a*  -  b  .  sin  ft"-  }*  -f  4  a  b  .  sin  a2  .  sin  /32 .  cos  Ce. 

1028.  If  7  be  the  angle  contained  between  the  polarizing  axes,  since  n,  /3,  7  are  the  sides  of  a  spherical  triangle, 
and  C  the  angle  included  between  the  sides  a  and  /3,  or  opposite  to  7,  we  have 

cos  a  .  cos  8  —  cos  7 

cos  C  = r—  -  , 

sin  a  .  sin  ft 

and  if  this  be  written  for  cos  C  in  the  latter  of  the  expressions  above  given  for  T-,  we  find  on  reduction 
T*  =  {  a  .  sin  a2  +  b  .  sin  £2  Js  —  4  a  b  {  1  —  cos  at  -  cos  ft"  -  cos  7*  +  2  .  cos  a  .  cos  ft  .  cos  7  }  . 

1029.  Carol.  If  the  polarizing  axes  be  at  right  angles  to  each  other,  7  =  90°  and  cos  7  =  0,  and  the  expression  tor 
the  compound  tint  becomes  T1  =  {  a  .  sin  a4  +  b  .  sin  p*  J*  —  4  a  b  (sin  aj  —  cos  ft1). 

1030  Proposition.   Two  rectangular  polarizing  ceres,  either  both  positive  or  both  negative,  being  given,  two  other  axes, 

or  fixed  lines,  may  be  found,  such  that  calling  0  and  O1  the  angles  made  with  them  respectively  by  a  ray  traversing 
a  spherical  portion  of  the  medium,  theAnt  polarized  shall  be  proportional  to  sin  0  .  sin  6'.* 

Resultant  Let  A  C  and  B  C  (fig.  199)  be  the  wo  polarizing  axes  including  a  right  angle,  of  which  let  B  C  be  the  more 
axes  arising  powerful.  Let  O  C  be  a  ray  penetrating  the  crystal  in  that  direction  ;  and  in  a  plane  P  C  Q  perpendicular  to 
from  the  A  C  B,  draw  any  two  lines  PC,  Q  C,  making  equal  angles  with  B  C,  either  of  which  we  will  represent  by  i. 
joint  action  Then  if  a  sphere  about  c  as  a  centre  be  conceived,  it  will  intersect  the  planes  AC  B,  P  C  Q,  OCA,  O  C  B, 
toJSiT"  O  C  P,  O  C  Q  in  lines  of  great  circles  B  A,  P  B  Q,  O  A,  O  B,  O  P,  O  Q,  and  we  shall  have  P  B  =  Q  B  =  T, 
polarizing  O  A  =  o,  O  B  =  ft,  O  P  =  0,  O  Q  =  O1 ;  and  by  Spherical  Trigonometry,  from  the  triangle  O  B  P,  we  have 
»xes,  ,.  s:n  o  A  \ 

Fig.  199-  cos  O  B  P  (  =  sin  O  B  A  =  sin  A  O  B  . : =  sin  a  .  sin  C,  since  A  B  =  90°  ) 

\  sin  A  B  / 

cos  ft  .  cos  x  —  cos  6 
sin  p  .  sin  x 

*  M.  Biot  appears  to  have  first  noticed  the  fact  announced  in  this  proposition,  vit.  that  Dr.  Brewster's  hypothesis  of  polarizing  axes  leads 
to  a  result  mathematically  identical  with  his  own  elegant  law  of  the  product  of  the  sines.  He  has,  however,  suppressed  his  demonstration. 
Dr.  Brewster's  verification  of  this  coincidence  of  results  seems  to  have  been  founded  on  a  numerical  comparison  of  Biot's  experiments  on 
•ulphate  of  lime  with  his  own  theory. 


LIGHT.  547 

Light,     and  therefore  —  cos  0  =  sin  a.  .  sin  /3  .  sin  *  .  sin  C  —  cos  /3  .  cos  x,  P»rt  IV. 

—  v—  '  and  similarly  from  the  triangle  O  B  Q,  since  O  B  Q  =  90°  -j-  O  B  A,  we  obtain  a  second  relation  s—v—  • 

-j-  cos  ff  =  sin  a  .  sin  ft  .  sin  x  .  sin  C  -(-  cos  /3  .  cos  x  ; 
and,  adding  and  subtracting,  (putting,  for  brevity's  sake,  cos  <X  =  p,  cos  a  =  7,) 

p  +  q  =  2  .  cos  /3  .  cos  ,r  ;        p  —  q  =  2  .  sin  a  .  sin  /3  .  sin  jc  .  sin  C. 

These  equations  express  the  geometrical  relations  subsisting  between  the  lines  P  C,  Q  C,  and  the  axes  A  C,  B  C  ; 
and,  if  combined  with  the  equations  of  Art.  1028  and  1029,  suffice  to  eliminate  a,  /3,  and  C,  and  to  express 
T  in  terms  of  x,  6,  and  &  alone.  To  execute  this,  we  have  by  the  equations  just  demonstrated 

P  +  1  V  =  cos  t?  ;        (£^-}*=  sin  a*  .  sin  ^  .  sin  C«  ; 
2.cos,r/  \2.sm;F/ 

and  in  the  latter,  putting  1  -  cos  C*  for  sin  C5;  and  for  cos  C2  its  value  given  by  Art.  1028,  which,  since  <y  = 
90°,  becomes  simply 

sin  o*  .  sin  /3s  .  cos  C«  =  cos  a*  .  cos  /3«, 

we  have  (  —  --  —  |  =  sin  o«  .  sin  /3*  -  cos  a.'  .  cos  /3*, 

\2  .  sin  x/ 

=  sin  o«  —  cos  jS9. 
Hence  we  get,  for  the  values  of  sin  d«  and  sin  /:?', 


2.cosjr/        \2  .  sm  x 


.  cos 
and,  substituting  these  in  the  equation  of  Art.  1029, 


(p  _ 

u 


.  . 

4  .  cos  a«  4  .  sin  .r2  I  sin  i2 

Such  is  the  general  form  of  the  expression  for  the  tint,  when  referred  to  arbitrary  axes  in  the  manner  here  sup- 
posed, and  it  is  complicated  enough  ;   but  if  we  fix  the  position  of  the  new  axes  so  as  to  make  sin  ct*  =  —  — 

the  complication  disappears  ;    we  have  then  --  —  =  —  ,  and  -  •  =   —  ,  so  that  the  value  of 

4  .  sm  x*  4  4  .  cos  j*  4 

T*   reduces  itself  to 


=  6'  {  (i  -  P  q)'  -  (p  -  ?)°  }  =  b'  {  i  -  if  -  q'  +  p*  9s  }  . 

=  6'  (1  —  ps)  (1  -  7s)  =  6s  .  sin  &'  .  sin  G", 

restoring  the  values  of  p  and  q,  or  cos  &'  and  cos  0,  consequently 

T  =  -  6  .  sin  e  .  sin  tf. 

The  negative  sign  is  prefixed  for  the  reason  stated  further  on  in  Art.  1034. 

Thus  we  see  that  the  combined  action  of  the  two  axes  in  the  manner  here  supposed,  on  Dr.  Brewster's  prin-      103  . 
ciples,  will  give  rise  to  a  series  of  isochromatic  lines  arranged  in  the  form  of  sphero-lemniscates  about  two  poles 
P,  Q,  determined  by  the  condition 

_  „        .    „  .    /  intensity  of  the  feebler  axis 

sin  B  P  =  sin  B  Q  =  \/  —  -  -  ; 

intensity  of  the  stronger 

and  'he  lines  C  P,  C  Q  so  determined  have  therefore  the  character  of  the  optic  axes  in  biaxal  crystals,  and  may 
be  designated  with  Dr.  Brewster  by  the  name  of  resultant  axes.  We  must  be  careful,  however,  not  to  confound 
a  resultant  with  a  polarizing  axis  in  this  theory. 

If  the  polarizing  axes  be  not  of  the  same  denomination,  as  if  one  be  positive  and  the  other  negative,  the  1032 
value  of  sin  B  P  becomes  imaginary,  and  the  tints  cannot  be  so  arranged.  But  if  we  suppose  the  new  axes  to  Combina" 
iie  in  this  case  in  the  same  plane  with  the  polarizing  ones,  as  in  fig.  200,  all  other  things  remaining,  we  tion  »f  » 
have  here  positive 

cos  O  B  A  =  +  cos  O  B  Q,         and  cos  O  B  A  =  -  cos  O  B  P, 


cos  a  cos  B  .  cos  x  —  cos  O1  axis. 

but  cos  O  B  A  =  --  :  —  —  ,         and  cos  O  B  Q  =  -   .    „  —  —  , 

sin  /3  sm  /3  .  sin  x 

so  that  we  find  cos  #'  =  p  =  cos  8  .  cos  x  -j-  cos  a,  .  sin  x  ; 

4  B  2 


548  LIGHT. 

Light,      and  similarly  cos  0  =  q  =  cos  j3  .  cos  x  —  cos  a  .  sin  x,  Part  IV 

"^v^~"'  whence,  by  adding  and  subtracting,  we  get  at  once  ^"  •*•/•»• 

p  —  q  p  +  9 

COS  a  =  -<—  -  ;-!-  ;  cos  /3  =    :f-  -  —  , 

2  .  sin  a;  2  .  cos  x 

which,  substituted  in  the  value  of  T8,  give 

=     (a  +  6)  -          -       + 


c  6  2  ab  (sin  a?8  —  cos  x1) 

-  4  a  6  +     ,  —  -  (p  «  +  o  9)  H  --  -  o  o. 

sin  x2  .  cos  x*  sin  x«  .  cos  x* 

Now,  if  in  this  we  suppose  —     --  1  --  -  =  0,  or  tan  x!  =  --  —  ,  it  will,  on  substitution  and  reduction,  take 

sin  -a1        cos  x*  •  b 

the  form 


COS  X*  COS  X* 

—  b 
and  T  =r  --  .  .  sin  0  .  sin  ff  ; 

COS  X* 

that  is,  restoring  the  value  of  x,  f  since  tan  x*  =  —    ,—  ,  and  therefore  cos  x8  =  -  —  -  1  ,  finally, 

T  =  -  (6  —  a)  .  sin  0  .  sin  0'. 

1033.  Thus,  in  this  case  also,  the  isochromatic  lines  are  sphero-lemniscates,  and  the  only  difference  is  that  their 
Position  of   poles  lie  now  in  the  plane  of  the  polarizing  axes,  instead  of  at  right  angles  to  it  ;  and  that  whereas  in   the 
lesultant  i  - 

this  case,     firmer  case  the  semi-angle  between  them  (x)  was  given  by  the  equation  sin  x  =    V/  —  ,  that  is,  cos  x  = 

\/  -      —  ,  in  this  it  is  given  by  the  equation  cos  x  —  y   -  —  —  . 

1034.  Carol.  1.  In  the  case  when  a  =  ft,  or  when  the  two  polarizing  axes  are  of  the  same  denomination  and  of  equal 
Cases  of  the  intensity,  we  have  sin  *  =1,  or  x  =  90°,  so  that  the  angle  between  the  resultant  axes  being  180°,  they  form  one 
resolution     straight  line,  the  lemniscates  become  circles,  and  the  single  resultant  axis  has  now  the  characters  of  a  polarizing 
axis  into       ax's-     Hence,  vice  versa,  a  single  polarizing  axis,  in  any  direction,  may  be  resolved  into  two  others  equal  in 
two.             intensity,  at  right  angles  to  it  and  to  each  other,  and  of  an  opposite  denomination  to  the  resolved  axis.     This 

follows  from  the  negative  sign  of  T,  which  is  prefixed  in  extracting  the  square  root  in  Art.  1030  and  1032  ; 
because  in  the  case  supposed,  when  the  arc  A  B  is  90°  the  angle  C  or  AO  B  is  necessarily  greater  than  90°,  and 
2  C  the  angle  of  the  parallelogram  of  tints  >  ISO0;  so  that  the  diagonal  will  be  to  be  measured  backwards 
through  the  angle,  or  must  be  a  negative  quantity. 

1035.  Carol.  2.   Since  a  single  axis  is  equivalent  to  two  equally  intense  axes  of  an  opposite  character  at  right  angles 
Composi-     to  it  and  to  each  other,  if  we  superadd  to  both  another  equal  axis  also  of  the  opposite  kind,  and  in  the  direction 
tion  of  three  of  tne  fjrst;  this  will  destroy  the  effect  of  the  first,  and  therefore  the  combination  of  three  equal  and  similar  axes 

?ula7        arising  on  the  other  side  at  right  angles  to  each   other,  will  be  equivalent  to  none   at  all.     Thus,  three  equal 

axes.  rectangular  axes  of  the  same  character  destroy  each  other's  effects.     This  is  Dr.  Brewster's  account  of  the  want 

of  polarization  and   double  refraction   in  crystals  whose  primitive  form  is  the  cube,  regular  octohedron,  &c., 

and  whose  secondary  forms  indicate  a  perfect  symmetry  in  their  molecules  with   respect  to   three  rectangular 

axes. 

1036.  There  is  no  necessity  to  pursue  further  the  general  subjects  of  this  species  of  composition  of  axes  and  of  tints, 
Indeed,  it  appears  to  us  that  the  rule  for  the  parallelogram  of  tints,  as  laid  down  by  Dr.  Brewster,  becomes 
inapplicable  when  a  third  axis  is  introduced  ;   for  this  obvious  reason,  that  when  we  would  combine  the  com- 
pound tint  arising  from  two  of  the  axes  (A,  B)  with  that  arising  from  the  action  of  the  third  (C,)  although  the 
sides  of  the  new  parallelogram  which  must  be  constructed  are  given,  (viz.  the  compound  tint  T,  and  the  simple 
tint  <",)  yet  the  wording  of  the  rule  leaves  us  completely  at  a  loss  what  to  consider  as  its  angle,  inasmuch  as  it 
assigns  no  single  line  which  can  be  combined  with  the  axis  C  in  the  manner  there  required,  or  which  quoad  hoi' 
is  to  be  taken  as  a  resultant  of  the  axes  A,  B.     For  further  information  therefore  on  this  subject  we  shall  content 
ourselves  with  referring  the  reader  to  his  original  Paper  in  the  Transactions  of  the  Royal  Society,  1818. 

§  X.     Of  Circular  Polarization. 

1037.  The  first  phenomena  referable  to  the  class  of  facts  to  whose  consideration  this  section  will  be  devoted,  were 
noticed  by  M.  Arago  i'n  his  Memoir  published  among  those  of  the  Institute  for  1811  on  the  colours  of  crystal- 
lized plates.     He  observed  that  when  a  polarized  ray  was  made  to  traverse  at  right  angles  a  plate  of  rock  crysta' 


LIGHT. 


Light,      (quartz)  cut  perpendicularly  to  the  axis  of  double  refraction,  on  analyzing  the  emergent  ray  by  a  doubly  refracting    Part  IV. 
""V*"1  prism,  the  two  images  had  complementary  colours,  and  that  these  colours  changed  when  the  doubly  refracting  •— -x— «. •' 
prism  was  made  to  revolve;    so  that  in  the  course  of  a  half  revolution,  the  extraordinary  image  (for  example)  Phenomena 
which  at  first  was  red,  became  in  succession  orange,  yellow,  yellow-green,  and  violet,  after  which  the  same  series  of  °o] j"^^,",, 
tints  would  of  course  recur.    It  is  evident  that  this  is  just  what  would  take  place,  supposing  the  several  coloured 
rays  at  their  emergence  from  the  rock  crystal  to  be  polarized  in  different  planes ;  and  to  this  conclusion  M.  Arago 
came  in  a  second  Paper,  subsequently  read  to  the  Institute.     The  subject  was  resumed  by  M.  Biot,  in  a  Paper 
published  in  the  Mem.  de  VTnst.,  1812;  and  his  labours  were  completed  in  a  second  extremely  interesting  Paper 
read  to  that  body  in  September,  1818. 

When  a  polarized  ray  is  made  to  traverse  the  axis  of  Iceland  spar,  beril,  and  other  uniaxal  crystals,  we  have      1038. 
seen  that  it  undergoes  no  change  or  modification ;  and  that  when  analyzed  at  its  egress  by  a  doubly  refracting  Rotatory 
prism,  having  its  principal  section  in  the  plane  of  primitive  polarization,  the  ordinary  image  will  contain  the  phenomena 
whole  ray,  or  the  complementary  tints  will  be  white  and  black.     Quartz,  however,  is  an  exception  to  this  rule.  °  ''"'" 
A  polarized  ray  transmitted,  however  precisely,  along  its  axis,  is  still  coloured  and  subdivided,  and  that  the  more, 
evidently,  the  thicker  is  the  plate.     If  we  place  on  a  proper  apparatus,  such  as  that  described  in  Art.  929  and 
figured  in  fig.  189,  a  very  thin  plate  of  this  body,  and  turn  round  the  analyzing  prism  M  in  its  cell,  till  the  extra- 
ordinary image  is  at  its  minimum  of  brightness,  it  will  in  this  position  have  a  sombre  violet,  or  purple  tinge, 
because  the  yellow  or  most  luminous  rays,  which  are  complementary  to  purple,  are  now  completely  extinguished. 
Let  the  angle  of  rotation  of  the  prism  in  its  cell,  measured  on  the  divided  circle  R,  and  which  in  this  case  will 
be  small,  be  noted  ;  and  then  let  the  rock  crystal  plate  be  detached,  and  another  cut  from  the  same  crystal,  but 
of  twice  the  thickness,  be  substituted.     The  tint  of  the  extraordinary  image  will  no  longer  be  violet ;   but  if  the 
prism  be  made  to  revolve  through  an  additional  equal  arc  in  the  same  direction,  the  violet  or  purple  tint  will  be 
restored,  and  the  minimum  of  brightness  attained ;    and,  in  general,  if  the  thickness  of  the  plate  (always  sup- 
posed cut  from  the  same  crystal)  be  greater  or  less  in  any  ratio,  the  angle  of  rotation  through  which  the  prism 
must  be  moved  in  the  same  direction,  to  produce  a  minimum  of  intensity  and  a  purple  tint  in  the  extraordinary 
image,  is  increased  or  diminished  in  the  same  ratio.     In  consequence,  if  the  plate  be  sufficiently  thick,  one  or 
more  circumferences  will  be  required  to  be  traversed ;  and  as  only  the  excesses  over  whole  circumferences  can  be 
read  off,  this  may  produce  some  confusion  or  doubt,  unless  we  take  care  to  use  a  succession  of  thicknesses  so 
gradually  increasing  as  not  to  allow  of  a  saltus  of  a  whole,  or  a  half  circumference. 

From  this  experiment  we  collect,  that  the  plane  of  polarization  of  a  mean  yellow  ray  which  has  traversed  the      1039. 
axis  of  a  quartz  plate,  has  been  turned  aside  from  its  original  position,  through  an  angle  proportional  to  the  Rotation 
thickness  of  the  plate  ;  and,  therefore,  assumes  at  its  egress  a  position  the  same  as  it  would  have,  had  it  revolved  °| the.  Plane 
uniformly  in  one  direction,  during  every  instant  of  the  ray's  progress  through  the  plate.     The  same  holds  good  °ioiy°  ' 
for  all  the  other  homogeneous  rays ;  but  to  prove  it,  we  must  abandon  the  use  of  white  light,  and  operate  with 
pure  rays  of  the  particular  colour  we  would  examine.    If  we  use  pure  red  light,  for  instance,  or  defend  the  eye  with 
a  pure  red  glass,  the  same  will  be  observed,  only  that  instead  of  a  violet  tint  and  a  minimum  of  light,  we  shall 
have  a  total  obliteration  of  the  extraordinary  pencil  when  the  prism  attains  its  proper  position,  thus  proving, 
what  in  the  former  mode  of  observation  might   have  been  doubtful,  that  the  polarization  of  the  emergent  ray 
is  complete. 

In,  examining  in  this  way  the  quantity  by  which  one  and  the  same  plate  of  quartz  turns  aside  the  planes  of     1040. 
polarization  of  the  different  homogeneous  rays,   M.  Biot  ascertained  that  the  more  refrangible  rays  are  more  Law  of  ro' 
energetically  acted  on  than  the  less,  and  have  their  planes   of  polarization  deviated  through   a   greater  arc.  t?''°? ol  th* 
According  to  this  eminent  philosopher,  the  constant  coefficient,  or  index,  which  represents  the  velocity  with  coloured 
which  the  plane  of  polarization  may  be  conceived  to  revolve,  is  proportional  to  the  square  of  the  length  of  an  rays. 
undulation  of  the  homogeneous  ray  under  consideration ;  so  that  if  we  call  X  the  length  of  an  undulation,  and 
t  the  thickness  of  the  plate,  the  deviation  produced  will  be  equal  to  k  .  X8  t,  k  being  a  certain  constant.     The 

18°.414 


value  of  this  constant  he  assigns  at 


when  t  is  reckoned  in  millimetres  ;  and  the  following  is  stated 


(6.18614)4' 

by  him  as  the  numerical  amount   of  the  deviations  in  degrees  (sexagesimal)  produced   by  one  millimetre  of 
thickness  of  rock  crystal  on  the  several  rays : 


Designation  of  the  homogeneous  ray. 


Arc  of  rotation  cor- 
responding to  one 
millimetre. 


Extreme  red     

Limit  of  red  and  orange 

Limit  of  orange  and  yellow   .  . 
Limit  of  yellow  and  green. . . . 
Limit  of  green  and  blue 
Limit  of  blue  and  indigo 
Limit  of  indigo,  and  violet.  . .  . 
Extreme  violet     .  .  . 


17°.4964 
20°.479S 
22°.3138 
25°.6752 
30°.0460 
34°.5717 
37°.6829 
44°.0827 


550  LIGHT. 

Light.          Jn  the  course  of  these  researches  M.  Biot  was  led  to  the  very  singular  discovery  of  a  constant  difference  sub-     Part  IV. 
—•••v""''  sisting  in  different  specimens  of  rock  crystal,  in  the  direction  in  which  this  rotation  or  angular  shifting  of  the   v>-"v""" 

1041.  plane  of  polarization  of  a  ray  traversing  them  takes  place.     In  some  specimens  it  is  observed  to  be  from  right 
j**,   to  left,  in  others  from  left  to  right.     To  conceive  this  distinction,  let  the  reader  take  a  common  cork-screw,  and, 

quartz  holding  it  with  the  head  towards  him,  let  him  turn  it  in  the  usual  manner,  as  if  to  penetrate  a  cork.  The  head 
will  then  turn  the  same  way  with  the  plane  of  polarization  of  a  ray  in  its  progress  from  the  spectator  through 
a  right-handed  crystal  may  be  conceived  to  do.  If  the  thread  of  the  cork-screw  were  reversed,  or  what  is  termed 
a  left-handed  thread,  then  the  motion  of  the  head  as  the  instrument  advanced  would  represent  that  of  the  plane 
of  polarization  in  a  left-handed  specimen  of  rock  crystal.  It  will  be  observed,  that  we  do  not  here  mean  to  say 
that  the  plane  of  polarization  does  so  revolve  in  the  interior  of  a  crystal,  but  that  the  ray  at  its  egress  presents 
the  same  phenomena  as  to  polarization  as  if  it  had  done  so.  This  is  necessary,  for  we  shall  see  presently  that 
a  very  different  view  of  the  subject  may  be  taken. 

1042.  In  crystals  which  present  this  remarkable  difference,  when  cut  and  polished,  and  when  the  external  indications 
Phenomena  of  crystalline  form  are  obliterated,  no  other  difference  can  be  detected.     Their  hardness,  transparency,  refractive 
draFcr^'slala  an(^  double  refractive  powers  are  the  same  ;  and,  with  the  exception  of  the  direction  in  which   it  takes  place, 

their  effects  in  deviating  the  planes  of  polarization  of  the  rays  which  traverse  them  are  alike.  Experiments 
subsequent  to  M.  Biot's  researches  have,  however,  established,  as  a  result  of  extensive  induction,  a  very  curious 
connection  between  this  direction  and  the  crystalline  forms  affected  by  individual  specimens.  In  the  variety  of 
crystallized  quartz,  termed  by  Hauy,  Plagiedral,  there  occur  faces  which  (unlike  those  in  all  the  more  common 
varieties)  are  unsymmetrically  related  to  the  axes  and  apices  of  the  primitive  form,  whether  regarded  as  the  rhomboid 
or  bipyramidal  dodecahedron.  Fig.  201  represents  such  a  crystal,  in  which  when  the  apex  A  is  set  upwards, 
the  faces  C,  C,  C,  are  observed  to  lean  all  in  one  direction,  viz.  to  the  right,  with  respect  to  the  axis,  as  if  dis- 
torted from  a  symmetrical  position  by  some  cause  acting  from  left  to  right  all  round  the  crystal.  When  the 
vertex  B  is  set  upwards,  the  same  distortion,  and  in  the  same  direction,  is  observed  in  the  plagiedral  faces 
D,  D,  D,  and  crystals  of  quartz  are  excessively  rare,  if  they  exist  at  all,  in  which  two  plagiedral  faces  leaning 
opposite  ways  occur.  Now  it  has  been  ascertained,  that  in  crystals  where  one  or  more  of  these  faces,  however 
minute  and  even  of  microscopic  dimensions,  can  be  seen,  we  may  thence  predict  with  certainty  the  direction  of 
rotation  in  a  plate  cut  from  it,  which  is  always  that  in  which  the  plagiedral  face  appears  to  lean  with  respect 
to  an  observer  regarding  it  as  the  reader  does  the  figure,  which  represents  a  right-handed  crystal.  Hence  we  are 
entitled  to  conclude,  that  whatever  be  the  cause  which  determines  the  direction  of  rotation,  the  same  has  acted  in 
determining  the  direction  of  the  plagiedral  faces.  Other  crystallized  minerals,  as  apatite,  &c.  also  present  pla- 
giedral and  unsymmetrical  faces ;  but,  independent  of  their  extreme  rarity,  they  are  not  possessed  of  the  property 
of  rotation  ;  so  that  at  present  we  are  unable  to  say  whether  this  curious  law  be  general,  or  to  conjecture  to  what 
principles  it  will  hereafter  prove  to  be  referable. 

1043.  When  two  plates  of  rock  crystal  are  superposed,  if  they  be  both  right-handed  or  both  left,  their  joint  rotatory 
Superposi-   effect  will  be  the  sum  of  their  respective  ones,  i.  e.  each  ray's  plane  of  polarization  will  be  shifted  through  an 
lion  of         a,,g.|e  equal  to  the  sum  of  those  through  which  it  would  have  been  shifted  by  their  separate  actions.     If  their 

characters  be  opposite,  it  will  be   their  difference,  i.  e.  the  index  of  rotation  in  a  right-handed  crystal  being 
crystal.         regarded  as  positive,  it  will  be  negative  in  a  left-handed  one. 

1044.  The  amethyst  (and,  possibly,  also  the  agate  in  some  cases)  presents  the  very  remarkable  and  curious  pheno- 
Amethyst.    menon  of  these  two  species  of  quartz  crystallized  together  in  alternate  layers  of  very  minute  thickness.     Accord- 
ingly, when  a  crystal  of  amethyst  is  cut  at  right  angles  to  the  axis,  and  examined  by  polarized  light  transmitted 
exactly  along  the  axis,  and  analyzed  as  usual,  it   offers  a  striped  or  fringed  appearance,   as   represented    in 
fig.  202,  variegated  with  different  colours,  according  to  the  different  planes  of  polarization  assumed  by  the  rays 
emergent  at  its  several  points,  and  presenting,  according  to  the  distribution  of  its  elements,  the  most  beautiful 
combinations  and  contrasts  of  coloured  fasciae  and  spaces.     For  a  particular  account  of  these  phenomena,  the 
reader  is  referred  to  a  Paper  by  Dr.  Brewster,  (Edinburgh  Transactions,  vol.  xi.)  who  first  observed  and  publicly 
described  them,  though  we  have  reason  to  believe  them  to  have  been  known  to  others  by  independent  observa- 
tion previous  to  the  publication  of  his  very  curious  and  interesting  Memoir.     The  layers  may  be  distinctly  seen 
cropping  out  to  the  surface  in  a  fresh  fracture  of  the  mineral,  and  imparting  that  peculiar  undulated  fracture 
which  is  the  chief  mineralogical  character  of  this  substance  by  which  it  is  known  from  ordinary  quartz. 

1045.  But  the  phenomena  of  rotation  as   above  described  are  not    confined  to  quartz.     Many  liquids,  and  even 
Rotatory      vapours  exhibit  it,  a  circumstance  which  would  seem  very  unexpected,  when  we    consider  .that    in  liquids   and 

£ases  tne  "»olec«les  must  be  supposed  unrelated  to  each  other  by  any  crystalline  arrangement,  and  independent 
of  each  other ;  so  that  to  produce  any  such  phenomena,  each  individual  molecule  must  be  conceived  as  unsym- 
metrically constituted,  i.  e.  as  having  a  right  and  a  left  side.  M.  Biot  and  Dr.  Seebeck  appear  about  the  same 
lime  to  have  made  this  singular  and  interesting  discovery ;  but  the  former  has  analyzed  the  phenomena  with 
particular  care,  and  it  is  from  his  Memoir  above  cited  that  we  extract  the  following  statements.  The  liquids  in 
which  he  observed  aright-handed  rotatory  property,  according  to  our  sense  of  the  word  above  explained,  in  which 
the  observer  is  supposed  to  look  in  the  direction  of  the  ray's  motion,  are  oil  of  turpentine,  oil  of  laurel,  vapour 
of  turpentine  oil,  and  an  alcoholic  solution  of  artificial  camphor  produced  by  the  action  of  muriatic  acid  on  oil 
of  turpentine.  The  left-handed  rotation  was  observed  by  him  in  oil  of  lemons,  syrup  of  cane  sugar,  and  alco- 
holic solution  of  natural  camphor.  In  all  these,  the  intensity  of  the  action,  or  the  velocity  of  rotation,  was 
much  inferior  to  quartz.  The  following  are  their  indices  of  rotation,  or  the  arcs  of  rotation  produced  by  one 
millimetre  of  thickness  in  the  plane  of  oolurization  of  a  certain  homogeneous  red  ray  chosen  by  M.  Biot  for  a 
standard,  as  calculated  from  his  data. 


L  I  G  II  T.  551 

Right-handed.  Imlex  of  rotation.  Left-handed.  Index  of  rotation.  Part  IV. 

Rock  crystal     -f  18°.414  Rock  crystal -  18°.414  v— v— ' 

Oil  of  turpentine     -f-    0°271  Oil  of  lemon    -     0°.436 

Ditto,  another  specimen -f-    0°.251  Concentrated  syrup  of  sugar     —     0°.5b4 

Ditto,  purified  by  repeated  distillations   -f     0°.286 
Oil  of  laurel 


Solution  of  1753  parts  of  artificial")        ,      ,,onlQ 
camphor  in  17359  of  alcohol    . .  j 


It  follows  further  from  M.  Biot's  researches,  that  when  any  two  or  more  liquids  are  mixed  together,  or  com-      1046. 
bined  with  plates  of  rock  crystal,  the  rotation  produced  by  the  compound  medium  will  be  always  the  sum  of  the  Law  of 
rotations  produced  by  the  several  simple  ones,  in  thicknesses  equal  to  their  actual  thicknesses  present  in  the  rotation  in 
combination,  the  thicknesses  in  mixed  liquids  being  assumed  in  the  ratio  of  the  volumes  of  each  respectively  Dllxlures- 
mixed;  so  that  calling  T  the  compound  thickness,  and  R  the  resulting  index  of  rotation,  we  shall  always  have 

R  .  T  =  r  .1+1*.?+  r" .  t"  -f  &c. 

where  r,  /,  &c.  are  the  indices  (with  their  signs)  of  the  elementary  ingredients,  and  f,  t',  &c.  their  thicknesses. 
Thus,  when  66  parts  by  measure  of  oil  of  turpentine,  having  the  index  +  0.253  are  made  to  act  against  38  of 
oil  of  lemon,  we  have 

-f-  66  X  0.251  —  38  X  0.436  =  0.002, 

so  that  these  thicknesses  ought  almost  exactly  to  compensate  each  other  ;  and  such  was,  in  fact,  the  result  ot 
M.  Biot's  experiment,  the  whole  pencil  transmitted  being  found  to  retain  its  primitive  polarization  without  the 
least  trace  of  an  extraordinary  image.  Again,  when  into  two  tubes  of  the  same  bore,  but  of  very  unequal 
lengths,  equal  quantities  of  oil  of  turpentine  were  poured,  and  the  rest  of  their  lengths  filled  with  sulphuric 
ether,  which  has  no  rotatory  property,  or  in  which  r  =  0,  the  two  compound  thicknesses  thus  differently  con- 
stituted gave  identically  the  same  tints  in  all  positions  of  the  analyzing  prism.  Thus  we  see  that  dilution  or 
mixture  which  only  separate,  without  decomposing  the  molecules,  do  not  alter  their  rotatory  power.  Nay,  even 
when  reduced  to  vapour,  M  Biot  found,  that  oil  of  turpentine  still  preserved  its  property  and  peculiar  character; 
and,  had  not  the  explosion  of  his  apparatus  prevented  accurate  measures,  would  probably  enough  have  been 
found  to  retain  the  same  index  of  rotation  allowing  for  the  change  of  density.  From  these  circumstances  he 
concludes  that  the  rotatory  power  is  essentially  inherent  in  the  molecules  of  bodies,  and  carried  with  them  into 
all  their  combinations.  But  this  is  too  rapid  a  generalization  ;  for  neither  sugar  nor  camphor  in  the  solid  state 
possess  this  property,  though  examined  for  it  in  the  same  circumstances  as  quartz  is,  by  transmitting  the  pola- 
rized ray  along  their  optic  axes;  and,  on  the  other  hand,  quartz  held  in  solution  by  potash,  or  (as  Dr.  Brewster 
has  found)  melted  by  heat,  and  thus  deprived  of  its  crystalline  arrangement,  manifests  no  such  property.  This 
obscure  part  of  chemical  optics  well  deserves  additional  attention. 

M.  Fresnel's  researches  have  been  directed  to  the  rotatory  phenomena  with  the  same  brilliant  success  which      1047. 
has  distinguished  his  other  inquiries  into  the  nature  of  light ;  and  he  has  shown  that  they  may  be  explained  by  Fresnel's 
conceiving  the  molecules  of  the  ether,  which  propagate  rays  along  the  axis  of  quartz,  or  rotatory  fluids,  instead  lheorv  ot 
of  vibrating  in  straight  lines,  to  revolve  uniformly  in  circles,  in  the  manner  explained  in  Art.  627,  (where  we  fj^Uo,!'0" 
have  shown  (Corol.)  that  such  a  mode  of  vibration  may  subsist,  and  must  arise  from  the  interference  of  two 
rectangular  vibrations  of  equal  amplitude,  but  differing  in  phase  by  a  quarter  undulation,)  and  by  admitting 
that,  in  virtue  of  some  peculiar  mechanism  in  the  molecules  of  the  media  in  question,  such  circular  vibrations, 
when  performed  from  right  to  left,  bring  into  play  an  elasticity  slightly  different  from  that  which  propagates 
them  forward  when  performed   in  the  contrary  direction.     The  colours  produced   by  such  media  he  conceives 
to  originate  in  the  interference  of  two  pencils  thus  circularly  polarized,  and  lagging  the  one  behind  the  other 
by  an  interval  of  retardation  proportioned  to  their  difference  of  velocities. 

But  to  make  this  last  hypothesis  admissible,  it  is  incumbent  on  us  to  show  that  the  phenomenon  which  neces-      1048. 
sarily  accompanies  a  difference  of  velocities,  viz.  a  bifurcation  of  the  pencil  in  the  act  of  refraction  at  oblique  Peculiar 
surfaces,  really  takes  place.     This  has  accordingly  been  shown  by  M.  Fresnel,  by  an  experiment  which,  though  <loul)le  re- 
of  great  delicacy,  is  decisive  and  satisfactory.     From  a  crystal  of  quartz  he  procured  to  be  cut  a  prism  having       j""  j 
its  refracting  angle  150°,  and  its  faces  equally  inclined  to  the  axis  ;   so  that  a  ray  traversing  it  internally  parallel  by0circu- 
to  its  axis  should  be  incident  at  equal  angles,  viz.  of  75°  on  either  face.     As  this  is  too  great  to  allow  of  the  larly  polari- 
ray's  egress,  he  cemented  on  the  surfaces  the  two  halves  of  another  precisely  similar  prism  cut  from  another  *'nfT  media. 
rock  crystal  of  an  opposite  rotatory  character.     Thus  in  fig.  203,  A  C  B  is  the  first  prism,  and  the  side  C  B  of  the 
second   prism  CB  E  being  cemented  on  to  C  B,  this   prism  is  bisected  by  the  plane  B  D,  and  the  half  of  it 
D  B  E  transferred  to  the  other  side,  and  cemented  with  its  side  B  C  in  contact  with  A  C,  thus  producing  the 
achromatic  parallelepiped  F  A  B  D  ;    so  that  if  a  ray  be  incident  on  Q  in  the  direction  P  Q  parallel  to  the  base 
A  B,  i.  e.  to  the  axis  of  the  two  crystals,  it  will  traverse  all  three  in  the  direction  of  the  axes  of  their  spheroids 
of  double  refraction  ;  and,  therefore,  so  far  as  the  Huygenian  law  of  double  refraction  is  concerned,  ought  to 
undergo  no  division.     Now  it  is  evident,  that  if  the  ray  PQ  be  at  its  entry  into  AFC  divided  into  two  circularly 
polarized  in  opposite  directions,  the  one   (R)   moving  quicker  than  the  other  (L,)  then,  at  quitting  the  surface 
A  C,  a  bifurcation  must  take  place,  the  ray  R  being  least,  and  L  most  refracted.     In  this  state  they  are  incident 
on  the  medium  A  C  B,  and  now  the  portions  R  and  L,  by  reason  of  the  opposite  nature  of  the  media,  exchange 
velocities  ;  so  that  R,  which  at  its  emergence  from  the  fee*.  AC  of  F  A  C  was  least  refracted  upwards,  will  now 


LIGHT. 


Li'ht. 


1049. 

Characters 
of  circular 
polarization 


1030. 

Other  cha- 
racters of 
circularly 
polarized 
rays. 


1051. 


•1052. 

Another 
mode  of 
producing 
circular  po- 
larization. 


1053. 


1054. 


lOJj. 
Tints  pro- 
duced by 
circularly 
polarized 
up. 


be  most  refracted  downwards ;  and  thus  the  separation  of  the  images  will  be  doubled,  and  the  same  will  take 
place  at  the  common  face  C  B.  Thus  this  combination,  both  from  the  doubling  of  the  separation,  and  the 
greatness  of  the  angles  of  incidence,  is  peculiarly  well  adapted  to  render  sensible  any  bifurcation,  or  difference 
of  velocities,  however  small,  which  may  exist  along  the  axis.  Accordingly,  with  the  compound  prism,  so  con- 
structed, a  double  refraction  is  produced  ;  and  the  two  rays  are  really  observed  to  emerge,  making  a  sensible 
ingle  with  each  other. 

But  it  is,  moreover,  observed,  that  though  thus  separated  by  a  real  double  refraction,  the  two  pencils  have  not 
acquired  the  characters  which  double  refraction  usually  impresses  on  the  ordinary  and  extraordinary  rays,  at 
their  emergence,  but  very  different  ones.  In  common  cases  of  double  refraction  the  two  emergent  pencils  are 
each  wholly  polarized  in  opposite  planes,  and  either  of  them  when  examined  with  a  doubly  refracting  prism 
gives  two  unequal  images,  one  alternately  more  and  less  bright  than  the  other,  as  the  prism  revolves  through 
successive  quadrants.  This  is  not  the  case  with  the  two  pencils  produced  in  the  case  before  us,  for 

First,  Either  of  them,  when  examined  with  a  doubly  refracting  prism,  gives  constantly  two  images  of  equal 
intensity,  in  whatever  plane  the  principal  section  of  the  latter  be  placed.  In  this  respect,  then,  they  present  the 
characters  of  unpolarized  light,  and  may  be  regarded  as  each  consisting  of  two  rays  polarized  at  right  angles  to 
each  other.  But 

Secondly,  They  differ  from  ordinary,  or  unpolarized  light,  in  a  very  remarkable  property,  which  was  first 
discovered  by  Fresnel,  and  is  a  chief  distinctive  character  of  this  kind  of  polarization.  Suppose  either  of  them 
to  be  incident  at  right  angles  on  the  surface  A  B  of  a  parallelepiped  of  crown  glass  of  the  refractive  index  1.51, 
having  its  angles  ABC  and  ADC  each  54J°,  it  will  then  be  totally  reflected  at  the  internal  surface  B  C  ;  and 
(if  the  parallelepiped  be  long  enough)  again  in  the  same  plane  at  the  opposite  surface  A  D,  and  will  emerge  at 
length  perpendicularly  through  the  surface  B  C.  But  the  emergent  ray,  instead  of  comporting  itself  as  ordinary 
light,  will  now  be  found  to  be  completely  polarized  in  a  plane  45°  inclined  to  that  in  which  the  reflections  were 
made,  whatever  may  have  been  the  position  of  that  plane.  If  both  the  pencils  be  treated  in  this  manner,  it  will 
be  found  that  the  one,  after  its  two  total  reflexions  will  assume  a  plane  of  polarization  45°  in  azimuth  to  the  right, 
and  the  other  45°  to  the  left  of  the  plane  of  the  reflexions. 

Thus  we  see  that  the  effect  of  double  refraction  along  the  axis  of  quartz  is  to  impress  on  either  of  the  emer- 
gent pencils  opposite  polarizations,  or  modifications,  of  a  nature  totally  distinct  from  that  given  to  a  ray  by 
ordinary  reflexion,  or  by  double  refraction  through  Iceland  spar,  &c. ;  and,  as  in  the  last  described  experiment, 
so  long  as  the  ray  enters  perpendicularly  into  the  first  surface  of  the  glass  parallelepiped,  it  is  indifferent  in  what 
plane  the  two  reflexions  are  operated,  and  since  when  presented  to  a  doubly  refracting  prism  in  any  plane  indif- 
ferently it  always  divides  itself  into  two  equal  pencils,  it  is  evident  that  the  ray  thus  modified  has  no  sides,  i.  e. 
no  particular  relations  to  certain  regions  of  space ;  and  therefore  that  the  epithet  circular  polarization,  apart 
from  all  theoretical  considerations,  may  be  naturally  applied  to  this  peculiar  modification.  But  the  characters 
above  described  are  not  the  only  ones  belonging  to  a  ray  thus  modified,  for 

Thirdly,  Such  a  ray  being  transmitted  through  a  thin  crystallized  lamina,  and  parallel  to  its  axis,  is  divided 
by  subsequent  double  refraction  into  two  rays  of  complementary  colours,  thus  marking  a  decided  difference 
between  it  and  a  ray  of  common  light ;  while,  on  the  other  hand,  these  colours  are  not  the  same  with  those 
which  would  arise  from  a  ray  of  light  polarized  in  the  usual  way  and  similarly  analyzed,  but  differ  from  them 
by  an  exact  quarter  of  a  tint,  either  in  excess  or  defect,  as  the  case  may  be. 

Fourthly,  A  ray  so  modified  by  this  peculiar  double  refraction,  when  transmitted  again  along  the  axis  of 
rock  crystal,  or  through  columns  of  oil  of  turpentine,  of  lemons,  &c.,  and  then  analyzed  by  a  double 
refracting  prism,  gives  rise  to  no  phenomena  of  colour,  differing  in  this  from  polarized,  and  agreeing  with 
common  light. 

Another  independent  mode  of  impressing  on  a  ray  all  this  assemblage  of  characters  has  been  discovered  by 
M.  Fresnel.  It  consists  in  inverting  the  process  described  in  Art.  1049.  Thus,  into  the  side  CD  of  the  glass 
parallelepiped  there  mentioned,  let  a  common  polarized  ray  be  introduced  at  a  perpendicular  incidence,  the 
parallelepiped  being  so  placed  that  the  plane  of  internal  reflexion  at  the  side  A  D  shall  be  45°  inclined  to  that 
of  its  primitive  polarization.  Then,  after  undergoing  two  total  internal  reflexions  at  G  and  F,  it  will  emerge  at 
E  deprived  of  its  characters  of  ordinary  polarization  and  endowed  with  those  of  circular,  and  being  no  way 
distinguishable  from  one  of  the  pencils  produced  by  double  refraction  along  the  axis  of  rock  crystal. 

It  remains  to  show,  however,  that  the  characters  here  described,  as  impressed  on  a  ray  by  transmission  along 
the  axis  of  rock  crystal,  are  really  those  which  ought  to  belong  to  a  ray  propagated  by  circular  vibrations.  And, 
first,  it  follows  from  Art.  627,  that  this  latter  ray  is  the  resultant  of  two  rays  polarized  at  right  angles,  and  dif- 
fering in  their  phases  by  a  quarter  undulation.  It  must,  therefore,  of  necessity  possess  the  first  character,  viz. 
that  of  division  into  two  equal  pencils  by  double  refraction  in  any  plane,  for  the  same  reason  that  unpolarized 
light  is  so  divided,  the  difference  of  phases  having  nothing  to  do  with  this  character. 

In  the  next  place,  a  ray  propagated  by  circular  vibrations  when  incident  on  rock  crystal  in  the  direction  of  the 
;ixis,  will  (by  hypothesis)  be  propagated  along  it  by  that  elasticity  which  is  due  to  the  direction  of  its  rotation, 
the  wave  then  will  enter  the  crystal  without  further  subdivision,  and  there  will  be  no  difference  of  paths,  or  inte*- 
tcring  rays  ;it  its  emergence ;  and,  of  course,  no  colours  produced  on  analyzing  by  double  refraction,  which  is 
another  of  the  characters  in  question. 

When  a  ray  propagated  by  circular  vibrations  is  incident  on  a  crystallized  lamina  it  may  be  regarded  as 
composed  of  two,  one  polarized  in  the  plane  of  the  principal  section,  the  other  at  right  angles  to  it,  of  equal 
intensity,  and  differing  in  phase  by  a  quarter  undulation.  Each  of  these  will  be  transmitted  unaltered,  and 
therefore  at  their  emergence  and  subsequent  analysis  will  comport  themselves  in  respect  of  their  interferences, 
just  as  would  do  the  two  portions  of  a  ray  primitively  polarized  in  azimuth  45°,  and  divided  into  two  by  the 


Par!  IV 


LIGHT.  553 

Light.      double  refraction  of  the  lamina,  provided  that  a  quarter  undulation  be  added  to  the  phase  of  one  of  these  latter     Part  IV. 
— -•/——•  rays.     Now  such  rays  will,  as  we  have  shown  at  length  in  Art.  969,  produce  by  the  interference  of  their  doubly  v^ */ 
retracted  portions,  the  ordinary  and  extraordinary  tints  due  to  the  interval  of  retardation  within  the  crystallized 
lamina.     Hence,  in  the  present  case,  the  tints  produced  will  be  those  due  to  that  interval,  plus  or  minus  the 
quarter  of  an  undulation  added  to,  or  subtracted  from,  the   phase  of  one  of  the   portions  ;    and,   consequently, 
will  differ  one-fourth  of  a  tint,  or  order,  from  that  which  would  arise  from  the  use  of  a  beam  of  ordinary  polarized 
light  incident  in  azimuth  45°  on  the  lamina. 

There  remains  but  one  more  character  of  the  rays  transmitted  along  the  axis  of  quartz,  which  we  must  show  1056. 
to  belong  to  a  ray  propagated  by  circular  vibrations,  viz.  that  described  in  Art.  1049.  But  in  order  to  this  it  M»difica- 
will  be  necessary  to  state  the  result  of  M.  Fresnel's  researches  on  the  modifications  which  light  undergoes  by  tlons  '!"" 
total  reflexion  in  the  interior  of  transparent  bodies.  light  by 

When  a  ray  polarized  in  any  azimuth  is  incident  on  a  reflecting  surface  which  reflects  the  whole  of  the  inci-  total 
dent  light,  if  we  decompose  it  into  two,  the  one  having  its  vibrations  performed  parallel,  and  the  other  perpen-  reflexion, 
dicular  to  the  surface,  and  regard  each  of  these  as  independent  of  the  other;  it  is  evident  that  the  reflexion  of 
these  portions  will  be  performed  under  very  different  circumstances,  the  ethereal  molecules  having  in  the  former 
case  to  glide  as  it  were  on  the  surface,  and  therefore  parallel  to  the  strata  in  which  their  density  is  constant,  while 
in  the  latter  each  molecule  in  the  act  of  vibration  will  pass  into  strata  of  variable  density.  The  reflexions 
therefore  will  be  performed  at  different  depths  in  the  two  cases  ;  and  from  this  cause  will  arise  a  difference  of 
route,  and  a  consequent  difference  of  phase  in  the  reflected  portions,  so  that  the  total  reflected  ray  will  no  longer 
be  capable  of  being  regarded  as  one  having  a  single  origin,  but  as  two  of  unequal  intensities,  oppositely  pola- 
rized, and  differing  in  phase  by  a  quantity  depending  on  the  angle  of  incidence  and  the  refractive  power  of  the 
medium.  From  peculiar  considerations,  of  a  delicate  nature,  and  depending  on  a  discussion  of  the  imaginary 
forms  assumed  by  the  general  expressions  for  the  intensity  of  a  ray  reflected  at  any  angle  (Art.  852)  when  applied 
to  the  case  of  total  reflexion,  M.  Fresnel  has  been  led  to  the  following  expression  for  the  difference  of  phases  (S) 
of  the  two  portions  in  question. 

2  f .  (sin  iy  -  fr«  +  1)  .  (sin  Q«  +  1 

0*a+  1)  (sini)*-  1 

where  fi  is  the  index  of  refraction,  and  i  the  angle  of  internal  incidence.  This  formula,  it  is  to  be  observed,  is 
given  by  him,  not  as  strictly  demonstrated,  but  merely  as  Highly  probable,  as  an  interpretation  of  the  analytical 
meaning  of  the  imaginary  formula  alluded  to.  The  mode  of  its  deduction  being,  however,  independent  of 
experiment,  and  entirely  a  priori,  it  is  clear  that  if  found  verified  by  careful  experiment  in  circumstances 
properly  varied,  it  may  be  received  as  a  physical  law,  like  any  other  result  of  the  same  kind.  Now  we  have 
already  seen,  that  in  the  case  of  crown  glass,  where  fi  =  1.51  and  i  =  54^°,  a  polarized  ray,  having  its  azimuth 
45°,  reckoned  from  the  plane  of  total  reflexion,  has  its  polarization  destroyed,  and  becomes  resolved  into  a  ray 
having  the  other  characters  of  a  resultant  from  two  differing  45°  in  p'hase,  by  two  total  reflexions  at  this  angle, 
(Art.  J056.)  But  if  in  the  above  formula  we  make  p.  =  1.51,  and  i  =  54°  37',  we  shall  find  S  =  45°,  and 
2  c  =  90D,  so  that  the  above  equation  is  verified  in  this  case.  M.  Fresnel  also  found  that  the  same  effect  was 
produced  by  three  reflexions  when  the  angle  of  incidence  was  69°  12',  and  by  four  when  74°  42',  both  agreeing 
with  the  formula  which  gives  in  the  former  case  J  =  -J  90°,  and  in  the  latter  X  =  £  90°,  for  the  difference  of  phase 
gained  or  lost  by  one  portion  on  the  other  at  each  reflexion.  Similar  verifications  were  obtained  by  performing 
two  reflexions  at  the  internal  surface  of  glass,  and  two  at  the  confines  of  glass  and  water  at  angles  of  68°  27'. 

It  appears,  then,  that  when  a  ray  polarized  in  azimuth  45°  undergoes  two   total  reflexions  at  the  angles,  and      |Q57 
in  the  manner  described,  it  becomes  circularly  polarized  ;  and  if  vice  versd,  the  two  elements  of  a  ray  so  circu-  Explana- 
larly  polarized  be  made  to  retrace  their  course,  they  will   reunite   into  a  ray  polarized  completely  in  one  plane,  tion  of  the 
Tims  we  see,  that  all  the  characters  of  the  rays  transmitted  along  the  axis  of  rock  crystal  agree  with  those  of  a  rotatory 
ray  so  compounded,  and  possessing  circular  polarization.      In  order,  then,  to  explain  the  phenomena  presented  P 
by  a  polarized  ray  when  incident  on  a  plate  of  this  substance  cut  at  right  angles  to  its  axis,  we  must  first  regard 
the  ray  as  resolved  into  two  others  (which  we  will  call  A  and  B)  of  equal  intensity  ;  the  one  A  polarized  in  a 
plane  45°  inclined  to  the  right,  the  other  45°  inclined  to  the  left  of  the  vertical,  (which,  to  fix  our  ideas,  we  shall 
take  for  the  plane  of  primitive  polarization.)     Now,  since  by  Art.  615  a  ray  polarized   in  any  plane  may  be 
regarded   as   equivalent  to  two  rays  each  of  half  its  intensity,  differing  in  their  phases  by  a  quarter  undulation, 
let  us   conceive  the  ray  A  as  resolved  into   two,  A  a  polarized   in  the  plane   -4-  45°,   and    having  its  phase 
advanced  -f-  ^  undulation,  and  another  A  6  also  polarized  at  +   45°,  but  having  its   phase  retarded,  or  —  -J 
undulation,    so  that  A  a  and    A  b  differ   £   undulation  in    their   phases.      Similarly,    let    B  be    regarded    as 
decomposed  into  Ba  polarized   at  —  45°,  and  having  its   phase  +  ^undulation,  and  B6  polarized  also  at 
—   45°,   but  having  its  phase   —  -J   undulation   different   from   B.      Thus  will  the  original   ray   be   resolved 
into   the    four   A.  a,  A  b,   B  a,  B  b.      Now,  let  us    combine  these  two  and    two  in  a  cross    order,    then  A  a 
combined  with  B  b  will  be  equal  rays,  polarized  in  opposite  planes,  and  differing  £  undulation  in  their  phases, 
and  will  therefore  compose  one  circularly  polarized  ray,  in  which  the  rotation  is  from  right  to  left.     Similarly, 
the  pair  A  b,  B  a  will  compound  another  equally  intense  circularly  polarized  ray  having  its  rotation  the  contrary 
way.     Now  these  will  (ex  hypotheai)  be   transmitted   through  the  quartz    with  unequal  velocities,  and  thus  an 
interval  of  retardation  will  arise,  and  if  the  surface  of  egress  or  ingress  be  oblique  to  the  axis,  a  double  refrac- 
tion will  take  place;  and  two  circularly  polarized  rays  will  emerge  in  different  directions,  as  experiments  show 
they  do.     If  perpendicular  they  will  emerge  superposed,  and  will  compound  one  ray.     Let  us  now  examine  what 
will  be  the  character  and  slate  of  polarization  of  this  compound  ray.     To  this  end  conceive  a  molecule  of  ether  C 
to  be  at  once  agitated  by  two  circular  motions  in  opposite  directions;  one  in  a  circle  equal  and  similar  to  A  P  in 

vol.  iv.  4  c 


554  L  I  G  H  T. 

Light,  the  direction  A  P,  the  other  in  a  circle  equal  and  similar  to  B  Q,  and  in  the  direction  B  Q,  fig.  205.  Let  A,  B  Part  IV. 
— — v— ^  be  two  molecules  setting:  <>"t  at  once  from  A,  B  in  these  circles  with  equal  velocities,  then  will  the  motion 'of  — v— 
Fig.  206.  C  at  any  instant  be  equal  to  that  compounded  of  the  motions  of  A  and  B  at  that  instant.  When  A  comes  to  P 
let  B  come  to  Q,  then  arc  AP=  BQ,  and  the  motions  at  P  and  Q  will  be  each  resolved  into  two,  those  of  which 
parallel  to  C  D  (a  perpendicular  to  P  Q)  conspire,  while  those  in  the  directions  P  D  and  Q  D  parallel  to  P  Q 
oppose,  and  being  equal  destroy  each  other ;  thus  C  will  move  only  in  virtue  of  the  sum  of  the  two  former,  and 
its  vibrations  will  therefore  be  rectilinear,  and  in  the  plane  C  D  perpendicular  to  P  D  Q.  If  the  thickness  of 
the  plate  of  quartz  were  nothing:,  or  such  that  the  interval  of  retardation  were  an  exact  number  of  undulations, 
A,  B  would  lie  at  opposite  extremities  of  a  diameter,  and  C  D  the  new  plane  of  polarization  would  be  per- 
pendicular to  AM  that  diameter,  or  coincident  with  the  plane  of  primitive  polarization.  But  if  not,  the  quicker 
motion  will  have  gained  on  the  other  a  part  of  a  circumference  M  B,  which  is  to  a  whole  circumference  as  the 
thickness  of  the  plate  is  to  that  which  would  produce  a  difference  of  a  whole  undulation  ;  and  at  the  emergence 
of  the  two  waves  into  air,  after  which  they  circulate  with  equal  velocity,  if  we  suppose  the  one  molecule  to  be 
setting  out  from  A,  the  other  will  be  setting  out,  not  from  M  the  opposite  extremity  of  the  diameter,  but  from  B, 
and  therefore  CD  the  new  plane  of  polarization  (which  from  what  has  just  been  shown  must  always  bisect  the 
angle  A  C  B)  will  no  longer  be  coincident  with  C  N  the  primitive  plane  of  polarization,  at  right  angles  to  A  M, 
but  will  make  an  angle  I)  C  N  with  it  equal  to  half  B  C  M,  and  therefore  proportional  to  M  B,  or  to  the  interval 
of  retardation,  i.  e.  to  the  thickness  of  the  plate.  Thus  the  system  of  rays  emerging  from  the  rock  crystal  plate 
will  compound  one  ray  polarized  in  one  plane,  and  in  the  position  the  original  plane  would  have  had,  had  it  revolved 
uniformly  round  the  ray  as  an  axis  during  its  passage  through  the  plate.  Thus  we  have  a  complete  and  satis- 
factory explanation  of  the  apparent  rotation  of  the  plane  of  polarization,  as  observed  by  Biot  in  the  case  of  a 
homogeneous  ray. 

1058.  It  is  observed,  that  the  spectra  formed  by  the  double  refraction  of  rock  crystal  along  its  axis  are  very  highly  and 
unequally  coloured.    The  violet  rays  are  most  separated,  and  therefore  the  difference  of  velocities  of  the  two  rotating 
pencils  is  much  greater  for  violet  than  for  red  rays.     Consequently,  the  apparent  velocity   of  rotation  of  the 
plane  of  polarization  will  also  be   greater  for  the  violet  rays  in   the  same  proportion,  and  thus  arise  all  the 
phenomena  of  coloration  observed  and  described   by  M.  Biot.     It  is  scarcely  possible  to  imagine  an  analysis 
of  a   natural    phenomenon   more    complete,   satisfactory,    and  elegant.      With   regard  to    the  physical  reason 
of  the  difference  of  velocity  in  the  two  circular  polarized  pencils  within  the  quartz,  it  is  true  we  remain  in  the 
dark;    but  the  fact  of  such  difference  existing  is  now  shown  to  be  no  hypothesis,  but  a  fact  demonstrated  by 
their  observed  difference  of  refraction,  and  by  the  observed  characters  of  the  two  emergent  rays. 

§  XI.   Of  the  Absorption  of  Light  by  Crystallized  Media. 

1059.  Crystallized   media,   endowed   with    the  property  of  double   refraction,  are   found  to  absorb  the   differently 
Absorption   coloured  rays  differently,  according  to  their  planes  of  polarization,  and  the   manner  in  which   these  planes  are 
r  h" 'uirize    presented  to  the  axis  of  the  crystal,  and  also  to  exert  very  different  absolute  absorbing  energies  on  rays  of  one 
double  re-    co'our  polarized  in  different  planes.     A  remarkable  instance  of  this  has  been  already  often   referred  to  in  the 
fracting        case  of  the  brown  tourmaline,  a  plate  of  which,  cut  parallel  to  the  axis,  absorbs  almost  entirely  all  rays  polarized 
crystals.       in  the  plane  of  the  principal  section,  and  lets  pass  only  such  among  oppositely  polarized  rays  as  go   to   con- 
stitute a  brown  colour. 

1060.  When  such  a  plate,  then,  is  exposed  to  natural  light,  since  at  the  entrance  of  each  ray  into  its  substance  it  is 
Properly  of  resolved  into  two,  one  polarized  in  the  plane  of  the  principal  section,  and  one  perpendicular  to  it,  the  former  is 

lin"  absorbed  in  its  progress  by  the  action  of  the  crystal,  while  the  brown  portion  of  the  latter  escaping  absorption, 
but  retaining  at  its  egress  the  polarization  impressed  on  it,  after  traversing  the  plate,  appears  with  its  proper 
colour,  and  wholly  polarized  in  a  plane  at  right  angles  to  the  axis.  Thus  the  curious  phenomenon  of  the  pola- 
Explained.  rization  of  light  by  transmission  through  a  plate  of  tourmaline,  or  other  coloured  crystal,  is  explained,  or  at 
least  resolved  into  the  more  general  fact  of  an  absorbing  energy  varying  with  the  internal  position  of  the  plane 
of  polarization.  The  crystal,  in  virtue  of  its  double  refractive  property,  divides  the  ray  into  two,  and  polarizes 
them  oppositely  ;  and  the  unequal  absorption  of  these  two  portions  mbtetpimtk)  causes  the  total  suppression  of 
one,  and  the  partial  of  the  other  of  the  portions  so  separated.  Thus  we  see  that  the  polarized  beam  obtained 
by  transmission  through  a  tourmaline  must  always  be  of  much  less  than  half  the  intensity  of  the  incident  light. 

1061.  The  destruction  of  the  pencil  polarized  in  the  principal  section  is   not,  however,  sudden;  for  if  the  plate  of 
Gradual       tourmaline  be  very  thin,  the  emerging  pencil  will   only  be  partially  polarized,  indicating  the  existence   in   it  of 

llou  rays  belonging  to  the  other  pencil.     This  is  best  shown   by  cult  in;.;-  a  tourmaline  into  a  prism  having  its  refract- 

ordinary       '"§'  edge  parallel  to  the   axis,  and  its  angle  small,  so  as  to  produce  a  wedge  whose  thickness  increases  not  too 

ray.  rapidly.     If  we  look  through   this  at  a  distant  candle,  we  shall   see   only  one  image,  viz.  the  extraordinary 

through  the  back  of  the  wedge,  (if  thick  enough  ;)  but  as  the  eye  approaches  the  edge,  the  ordinary  image  appears 

at  first  very  faint,  but  increasing  in  intensity  till,  at  the  very  edge,   it  becomes  equal  to  the  other.     At  the  same 

time  the  colour  of  the  latter,  which  at  first  was  intense,   becomes  diluted;   and  the  images  approximate  not  only 

to  equality  of  light,  but  to  similarity  of  tint.     We  see  by  this,  too,  that  in  strictness  the  ordinary  pencil  is  never 

completely  absorbed  by  any  thickness,  however  great ;  but  as  it  diminishes  in   geometrical   progression  as  the 

thickness  increases  in  arithmetical,  the  absorption  may  for  all  practical  purposes  be  regarded  as  total  at  moderate 

thicknesses 


L  I  a  H  T.  555 

Light.  The  indefatigable  scrutiny  of  Dr.  Brewstcr,  to  whom  we  owe  nearly  all  our  knowledge  on  this  subject,  has     par[  jy 

— v™"''  shown  that  the  same  property  is  possessed  in  greater  or  less  perfection  by  the  greater  number  of  coloured  doubly  >.__     ___. 
refracting  media ;  and  the  expression  of  the  property  may  be  rendered  general  by  considering  all  doubly  refrac-       1062. 
tive  media  as  possessing  two  distinct  absorbing  powers  or  two  separate  scales  of  absorption  for  the  two  pencils,  Media  pos- 
or  (adopting  the  language  of  §  III.  part  2)  as  having  two  distinct  types,  or  curves  expressing  the  law  of  absorp-  sess  two 
tion  throughout  the  spectrum.     If  these  types  be  both  straight  lines  parallel  to  the  abscissa,  tit"  "-"stal  will  be  !J{^"{:[ 
colourless.     Such  are  limpid  carbonate  of  lime,  quartz,  nitre,  &c.     If  they  be  similar  and  equal  em"ves,  the  powers'.05 
medium,  although  coloured,  will  present  the  same  colour,  and  the  same  intensity  of  tint,  in  common  as  in  pola- 
rized light.      If  dissimilar,  or  if,  although  similar,  their  ordinates  are  in  a  ratio  of  inequality,  the   character,  in 
the  former  case,  and  the  intensity  in  the  latter,  will  vary  on  a  variation  of  the  plane  of  polarization  of  the  inci- 
dent beam  ,  so  that  if  a  plate  cut  from  such  a  crystal  be  exposed  to  a  beam  of  polarized  white  light,  and  turned 
round  in  its  own  plane,  or  otherwise  inclined   to   the  beam,  its  colour  will  change  either  in  hue  or  depth  or 
both.     Dr.  Brewster  has  remarked   such    change   of  colour  and  the  phenomena  connected  with  it  in  a  great 
variety  of  crystals  both  wi,th  one  and  two  axes,  of  which  he  has  given  a  list  in  a  most  interesting  Paper  on  the 
the  subject  in  the  Philosophical  Transactions,  1819,  p.  1,  which  we  strongly  recommend  to  the  reader's  perusal. 
It  may  be  familiarly  seen  in  a  prism  of  smoked  quartz  of  a  pretty  deep  tinge,  which  held  with  its  axis  in  the 
plane  of  polarization  appears  of  a  purple  or  amethyst  colour,  while  if  held  in  a  direction  at  right  angles  to  this 
position,  its  colour  is  a  yellow  brown. 

But  in  order  to  analyze  the  phenomena  more  exactly,  we  must  examine  the  two  pencils  separately.     To  this       1063 
end  Dr.  Brewster  took  a  rhomboid  of  yellow  carbonate  of  lime  of  sufficient  thickness  to  give  two  distinct  images  Absorption 
of  a  small  circular  aperture  placed  close  before  it,  and  illuminated  with  white  light,  when  he  observed  that  the  of  the  rays 
image  seen  by  extraordinary  refraction  appeared  of  a  deeper  colour  and  less  luminous  than  the  other,  being  an  '"  thfj two 
orange  yellow,  while  the  ordinary  image  was  a  yellowish  v.'hite.     He  found,  moreover,  that  the  difference  of  [^"j^  *•*" 
colour  was  greater  as  the  paths  of  the  refracted  rays  within  the  crystal  were  more  inclined  to  the  axis,  being  0  crystals  with 
when  the   rays  passed  along  the  axis,  and  a  maximum  when  at  right  angles  to  it.     If  we  denote  by  ¥„  and  Ye  one  axis, 
the  ordinates  of  the  curves,  expressing  the  law  of  absorption  as  in  Art.  490,  for  the  ordinary  and  extraordinary 
pencil  respectively,  these  will  both  therefore  decrease  as  we  proceed  from  the  red  to  the  violet  end  of  the  spectrum, 
corresponding  to  types  of  the  character  of  that  represented  in  fig.  1 14  ;  but  Ye  being  smaller,  and  decreasing 
more  rapidly  than  Y0.     Moreover,  since  Y0  =  Y,  in  the  axis,  and   since  as  we  recede  from  the  axis  Y,,  increases  Foi-mulsefor 
(because  the  colour  of  the  ordinary  pencil  becomes  whiter  and  more  luminous)  while  Yc  diminishes  by  the  same  the  light 
degrees,  (the  extraordinary  becoming  deeper  and  less  bright,)  we  shall  represent   both  these  changes  satis-  transmitted 
factorily  by  putting 

Y0  =  Y  (1  -f  k  .  sin  0*)  ;         Yt  =  Y  (1  —  k .  sin  (ft). 

These  give  Y0  -f-  Y,  =  2  Y  =  constant,  or  independent  of  6,  which  agrees  with  an  observation  of  Dr.  Brewster, 
that  in  every  situation  the  combined  tints  of  the  two  images  are  exactly  the  same  with  the  natural  colour  of 
the  mineral,  (which,  in  this  instance,  appears  to  have  been  alike  in  all  directions.) 

In  this  case,  then,  the  colour  of  a  plate  of  the  crystal  of  given  thickness  exposed  to  natural  light  will  be  the       1064. 
same,  whether  the  plate  be  cut  parallel  or  perpendicular  to  the  axis.     But  Dr.  Brewster  has  observed,  that  this  Cases  of 
is  not  always  the  case,  but  that  great  differences  occasionally  exist  in  this  respect.     Thus  he  found,  that  in  some  two  distinct 
specimens  of  sapphire  the  colour  when  viewed  along  the  axis  was  deep  blue,  and  when  across  it  yellowish  green.  co'ours- 
In  Idocrase  an  orange-yellow  tint  is  seen  along  the  axis,  and  a  yellowish  green  across  it.     Specimens  of  tour- 
maline also  are  not  uncommon  in  which  the  tint  across  the  axis  is  green,  while  along  the  axis  it  is  deep  red  ; 
and,  in  general,  this  mineral  is  always  much  more  opaque  in  the  direction  of  the  axis  than  in  any  other;   so  much 
so,  indeed,  that  plates  of  a  very  moderate  thickness  cut  across  the  axis  are  nearly  impermeable  to  light.     One  of 
the  most  remarkable  instances  of  this  kind  we  have  met  with   is  a   variety  of  sub-oxysulphate  ot  iron,  which 
crystallizes  in  regular  hexagonal  prisms,  and  which  viewed  through  two  opposite  sides  of  the  prism   is  light 
green,  but   along  the  axis,  a  deep  blood  red,  so  intense  that  a  thickness  of  -fa  inch  allows  scarcely  any  light  to 
pass.     It  is  obvious,  that  to  such  cases  the  formulae  of  the  last  article  do  not  extend.     But  a  slight  modifi-  Investiga- 

cation  will  enable  us  to  embrace  the  phenomena  in  an  analytical  expression.     For  if  we  take  'lon  °; 

formula;  for 

y,  ~  X,  +  Y0  .  sin  &•,          y,  =  X.  +  Y.  .  sin  0* ;  these  cases. 

where  X0,  Y»,  &c.  as  well  as  y0,  y,  represent  functions  of  X  (the  length  of  an  undulation)  being  the  ordinates 
of  so  many  curves,  or  types  of  tints,  whose  relations  are  to  be  determined,  we  have 

2/0  +  y.  =  (X.  +  X.)  +  CY.  +  Y,)  sin  0\ 

Now  this  is  the  tint  which  a  sphere  of  the  medium  of  a  diameter  =  1  will  exhibit  when  viewed  by  natural  light 
along  a  diameter  inclined  0°  to  the  axis.  If  we  represent  by  A  and  B  the  ordinates  of  the  types  of  the  tints 
it  is  observed  to  exhibit  in  the  directions  of  the  axis,  and  perpendicular  to  it,  we  have,  when  0  ==  0, 

jf.-3-y.^AssX.-f  X.; 

and  when  6  =  90°, 

y.  +  y.  =  B  =  (X.  +  X.)  +  (Y.  +  Y.),  Expression 

whence  we  have  Y,  +  Y,  =  B  -  A ;  ^ 

and  the  tint  exhibited  by  ordinary  light  at  the  inclination  0  to  the  axis,  will  be  represented  by  transmitted 

rn          A  \         •      oo  in  COBlm°n 

ya  +  y,  =  A  +  (B  -  A)  .  sin  04,  Ught 

=  A  .  cos  0*  +  B  .  sin  0s. 

4  c  2 


556  LIGHT. 

Light     Thus  in  the  case  of  our  sub-oxysulphate  of  iron,  A  is  the  ordinate  of  the  type  of  a  deep  blood-red  tint,  and  B     Part  IV. 
—v—     in  like  manner  represents  a  bright  pale  green,  so  that  we  shall  have  at  any  intermediate  inclination  0  '— - -y^ 

tint  =  (deep  red)  X  cos  0*  -f  (light  green)   x  sin  0*, 

which  represents  faithfully  enough  the  gradual  passage  of  one  hue  into  the  other  as  the  inclination  changes 

Suppose  now  the  incident  beam  polarized  in  any  plane,  and  let  the  plane  in  which  the  ray  and  the  axis  of 
When  illu-  the  sphere  he  make  an  angle  =  a  with  that  plane.  Then  would  cos  «*  and  sin  «"-  represent  the  intensities  of 
>oTarLed  ^le  .orTary  and  extraordinary  pencils  which  superposed  make  up  the  emergent  beam,  were  the  crystal  limpid  • 
light  in  Vlrtue  0'  lts  absorbent  powers,  they  will  be  reduced  respectively  to 

ya  =  cos  a2  (X,  +  Y.  sin  0'-),      and  y,  —  sin  v?  (X,  -f  Y.  .  sin  0'), 

so  that  at  their  emergence  they  will  no  longer  make  up  white  light,  but  a  variable  tint  whose  type  has  for  its 
ordinate 

(X0 .  cos  «2  -f  X, .  sin  a"-)  -f-  (Y0 .  cos  «"-  -f-  Y,  .  sin  a"-)  .  sin  0*, 
in  which  it  will  be  recollected  that          X.  -f-  X.  =  A,  and  Y,  -)-  Y,  =  B  —  A. 

To  determine  the  individual  values  of  X0,  &c.  however,  we  must  have  two  more  conditions,  and  these  will  be 
found  by  considering,  first,  that  in  the  direction  of  the  axis  the  tint  must  be  independent  of  «,  which  gives 
X.  .  cos  a3  -f-  X. .  sin  a-  independent  of  a,  and  therefore  X0  =  X,,  and  either  of  them  =  A.  To  get  another 
condition,  let  the  tints  be  noticed  which  the  sphere  or  crystal  exhibits  when  its  axis  is  perpendicular  to  the  vis<iul 
ray ;  and,  first,  coincident  with,  next,  perpendicular  to,  the  plane  of  polarization,  i.  e.  when  a  =  0,  and  a  =  90°. 
These  are  respectively  X.  +  Y.,  and  X.  +  Y. ; 

and  calling  these  a  and  b,  we  have 

Y.  =  a  -  X0  =  a  -  A,     Y,  =  b  -  A. 
Hence  the  final  expression  for  the  tint  seen  in  polarized  light  will  be 

A  +  {  (o  —  A)  .  cos  a5  -)-  (6  —  A)  sin  a'2  }  .  sin  0\ 
that  is,  A  .  cos  6*  +  {  a  .  cos  «2  +  6  .  sin  «*  }  .  sin  G*, 

in  which  it  will  be  observed  that  a  and  6  are  complements  of  each  other  to  the  tint  B,  because 

a  +  6  =  X0  -f  Y.  +  X,  +  Y,  =  B,  by  Art.  1064. 

1066.  Such  is  the  expression  for  the  apparent  hue  of  crystals   with  one   axis,  which   exhibit  a  variable   colour  in 
Dichroism.  common  or  polarized  light,  according  to  their  position  with  respect  to  the  incident  light.     The  phenomenon  in 

question  may  be  generally  termed  dichroism,  though  the  word  has  usually  been  applied  only  to  that  particular 
case  where  a  marked  change  in  the  character  of  the  tint  takes  place,  as  from  red  to  green,  &c. 

1067.  The  dichroism  of  biaxal  crystals  differs  in  many  of  its  phenomena  from   those  having  only  one  optic   ;i\N. 
Dichroism    If  we  look  through  a  plate,  or  into  a  crystal  of  any  biaxal  mineral,  having  the  property  in  question,  illuminated 
in  biaxal      jjy  natural  light  in  such  a  direction  that  the  visual  r:iy  within  the  crystal  shall  pass  along,   and  in  the  immediate 
ip   one       neighbourhood  of,  one  of  the  axes,  we  shall  perceive  a  phenomenon  like  that  represented  in  fig.  206,  consisting 

of  two  similar  and  equal  sombre  spaces  A  B  one  on  either  side  of  the  pole  P,  and  of  the  principal  section  P  P', 
Colours  of  and  if  we  look  along  the  other  axis  P'  a  similar  pair  of  spaces  will  be  seen  in  its  neighbourhood.  In  the 
iolite.  mineral  called  dichroitt  by  Hauy,  (on  account  of  the  striking  difference  of  its  colours  in  different  positions,)  or 

iolitc  (from  its  violet  hue)  by  others,*  of  which  the  phenomena  have  been  described  by  Dr.  Brewster  in  the  Paper 
already  cited,  these  spaces  are  of  a  full  blue  colour,  while  the  intermediate  region  towards  O,  along  the  line  ()  P  C, 
Phenomena  and  the  space  beyond  P  towards  C  are  yellowish  white.     In  epidote  the  sombre  spaces  are  brown,  and  the  region 
of  epiclote.    around  O  and  in  the  principal  section  green,  of  a  greater  or  less  degree  of  dilution.     In  this  latter  minend  (at 
least  in  some  of  its  more  ordinary  varieties  of  crystalline  form,  vix.  in  long  striated  prisms  much  flattened,  and 
terminated  by  dihedral  summits  placed  obliquely,  so  as  to  truncate  two  of  the  angles  of  the  prism)  the  pheno- 
mena are  seen  without  any  artificial  section,  merely  by  looking  in  obliquely,  across  the  axis  of  the  prism  ;    and 
the  same  is  true   of  many  other  minerals,  as,  for  instance,  the  axinite,   in   which  the  transition  of   colour  is 
extremely  remarkable  and  beautiful 

1068.  The  phenomena  of  dichroism  in  biaxal,  as  well  as  in  uniaxal  crystals,  are  evidently  related  to  the  optic  axes, 
Connection  an(j  depend  on  the  planes  of  polarization  assumed  by  the  intromitted  light,  during  its  transit  through  the  crystal 

f  tliepbe-  t()  w|lose  absorptive  power  it  is  subjected.     Now,  if  we  consider  the  form  and  situation  of  the  sombre  spaces 
with  the       where  the  greatest  absorptive  energy  is  exerted,  we  are  at  once  struck  by  their  analogy  with  those  occupied  by 
polarized      the  more  vividly  coloured  parts  of  the  rays  about  the  axes  in  the  situation  of  fig.  179.     That  figure  represents 
rings  and      (Art.  900)  the  extraordinary  set  of  rings  as  seen  in  a  crystal  whose  principal  section  is  in  the  plane  of  primitive 
°Ptic  A*5S    polarization.     Fig.  207  represents  the  ordinary  or  complementary  set  as  seen  around  either  of  the  axes,  the 
pole  P,  and  the  principal  section  being  here  occupied  with  white  light,  and  very  bright,  in  consequence  of  its 
containing  the  whole  incident  light,  while  the  lateral  or  coloured  portions  occupied  by  the  rings  are  less  illu- 
minated, the  colours  originating  in  an  abstraction  of  certain  rays. 

Conceive  now  a  number  of  such  sets  of  coloured  rings  not  all  of  exactly  the  same  dimensions,  nor  having 

*  Mohs,  with  his  usual  contemptuous  disregard  of,  or  rather  hostility  to,  all  ordinary  convenience  and  received  usage,  chooses  to  call  this 
mineral  "prismatic  quartz."  Such  a  nomenclature  must  ere  long  work  out  ils  own  destruction,  but  while  it  subsists  the  nuisance  is  intolerable. 
We  cannot  but  lament,  that  such  a  cause  should  exist  to  raise  up  prejudice  against  a  system  in  many  respects  so  useful  and  valuable. 


LIGHT.  557 

precisely  the  same  pole,  but  very  nearly  so,  to  be  superposed  on  one  another,  then  would  the  colours  be  obliterated    Part  IV. 
•  and  blended  into  white  light  by  their  overlapping,  but  still  the  general  intensity  of  the  light  in  the  lateral  regions  ^—  •  -v—  ^' 
would  remain  much  feebler  than  in  the  principal  section,  and  the  effect  would  be  precisely  that  of  fi;T.  206,  viz.  Analogy  in 
two  sombre,  cloudy,  fan-shaped  spaces  traversed  by  a  narrow  ray  of  vivid  light,  opening  out  from  P  towards  C  rosPec.t  of 
and  O.      Such  would  be  the  case  with  a  limpid  crystal,  supposing  such  a  slight  degree  of  confusion  of  structure  'jj'6" 
as  to  produce  the  non-coincidence  of  the  rays  from  all  its  molecules.     In  this  case,  however,  neither  of  the  spaces 
in  question  would  appear  coloured,  nor  would  the  phenomena  be  seen  at  all   without  the  use  of  polarized  light 
and  its  subsequent  analysis.     I5ut   if  we  conceive   the  crystal,  instead  of  limpidity,  to  possess  the  property  of 
double  absorption,  the  suppressed  and  transmitted  portions  will  be,  not  white  light,  but  light  of  the  colour  of 
one  or  other  of  the  pencils  into  which  it  is  resolved  by  double  refraction,  according  to  its  plane  of  polarization 
and   the  thickness  of  the   medium  it  has  traversed  ;   and  the  analysis  of  the  emergent  ray  may  be  regarded  as 
performed,  at  least  imperfectly  by  the  difference  of  absorptive  powers  acting  differently  on  the  two  pencils.     In 
support  of  this  it  may  be  noticed,  that  when  we  examine  the  system  of  rings  in  the  usual  way,  by  polarized 
light,   in   crystals  presenting  the  above  phenomenon,  they  are  usually  found  to  be  very  irregular,  several  sets 
evidently  overlapping  and  interfering  with  one  another,  and  rendering  the  non-coincidence  of  all  the  axes  a 
matter  of  ocular  demonstration. 

In  Art.  931  we  investigated  the  law  of  intensity  of  the  illumination  of  the  polarized  rings  in  different  parts  of     1069. 
their  periphery  for  nniaxal  crystals.     As  what  is  there  said  does  not  apply,  however,  to  biaxal  ones,  and  as  the  DIGRESSION 
present  subject  has  led  us  to  the  consideration  of  the  more  general  case,  it  will  not  be  irrelevant,  if  we  digress  Theory  of 
at  this  point,  in  order  to  show,  what  modifications  the  statement  there  .made  must  receive  to  embrace  the  phe-  rizeoringi 
nomena  of  biaxal  crystals.  resumed. 

M.  Biot  has  stated  the  general  law  of  polarization  in  biaxal  crystals,  from  his  elaborate  researches  on  that  1070. 
subject  (Mem.  sur  les  Lois  Generates  de  la  Double  Refraction  et  Polarisation,  fyc.  Mum.  Acad.  Scl.  1819)  to  Blot's  ge 
be  as  follows  :  "eral  law 


If  two  planes  be  drawn  through  the  course  of  a  ray  within  a  crystal  and  through  the  two  optic  axes,  and  a  ^znes  of 
third  plane  bisecting  the  angle  included  between  the  two  former,  this  will  be  the  plane  of  polarization  if  the  ray  polarization 
be  an  ordinary  one  —  but  one  perpendicular  to  it  if  extraordinary.     Thus  in   fig.  209,  C  P  and  C  P'  being  the  in  biaxal 
optic  axes,  and   AC   a  ray  penetrating  the  crystal,  if  PA,  P'  A   be  joined   by  arcs   of  circles   on  the   sphere  Cl7stills' 
II  O  K  A  having  C  for  its  centre,  and  the  angle  PAP'  be  bisected  by  the  arc  A  N,  the  plane  A  C  N  bisecting 
the  dihedral  angle  between  the  planes  P  C  A  and  P'  C  A  is  the  plane  of  ordinary  polarization,  and  a  plane  per- 
pendicular to  it  that  of  extraordinary.     This  is  the  law  of  fixed  polarization,  and  expresses  generally  the  planes 
of  polarization  assumed  by  the  two  rays  at  their  emergence  from  doubly  refracting  crystals.     It  is  a  consequence 
of  FresnePs  general  theory,  (though   deducible  fro;n   it  by  a  train  of  analytical  reasoning  far  too  intricate  and 
refined  to  allow  of  its  insertion  in  a  treatise  like  the  present,)  and,  having  been  experimentally  established  lotif 
before  that  theory  was  devised,  must  be  looked  on  as  a  strong  additional  proof  of  its  conformity  to  nature. 

The  doctrine  of  movable  polarization,  however,  which,  so  far  as  respects  the  phenomena  of  the  colours  and      1071. 
intensity  of  the  rings,  has  been  shown   by  M.  Biot  in  the  same  excellent  paper,  to  represent  with  fidelity  their  Doctrine  of 
various  affections,  whether  in  uniaxal  or  biaxal  crystals,  requires  the  resulting  ray  to  assume  at  its  emergence  a  movable 
plane  of  polarization  alternately  coincident  with,  and  making  with  the  primitive  plane  of  polarization  twice  the  Pola';lzjltl0" 


angle  which  the   plane  of  fixed   polarization  so  determined  would  make  ;    so   that  if  we   draw  A  M   (fig.  208)  bfj^t  ,r'° 
bisecting  the  angle  PAP',  the  emergent  ray  will  be  affected  by  subsequent  analysis,  as  if  polarized  either  in  the  tals. 
plane  of  primitive  polarization,  or  making  with  it  an  angle  equal   to  twice  C  M  A,  and   from  this  it  is  easy  to  Fifj-  208. 
derive  the  law  of  intensity  in  question,  for  the  ray  by  which  the  point  A  of  the  rings  is  formed  consists  of  two  Law  of 
portions,  of  which  (A)  is  affected  by  subsequent  analysis  by  a  prism  of  Iceland  spar,  as  if  it  were  polarized  in  a  intensity  of 
plane  making  an  angle  2  C  M  A  =  ty  with  the  plane  of  primitive  polarization,  in  which  we  suppose  the  principal  l.hc  rin?s  '" 
section  of  the  analyzing  prism  to  be  placed,  and  the  other,  complementary  to  this  (1  —  A)   retains  its  primitive    |,j*.7of 
polarization.     The   portion  A  then   will   be  divided  between  the  ordinary  and  extraordinary  image   in  the  pro-  their  peri 
:  (sin  2  ty)-,  and  (considering  only  the  latter,)  A  being  its  intensity  at  its  emergence  from  the  phery. 


portion  (cos 

crystal,  A  .  (sin  2  Y')2  will  be  its  intensity  in  the  extraordinary  image,  or  in  the  primary  set  of  rings,  while  the 
whole  of  the  portion  1  —  A  will  pass  into  the  ordinary  or  complementary  set,  as  in  Art.  932,  so  that  we  have 
only  to  express  this  in  terms  of  the  azimuth  of  the  crystallized  plate  itself,  and  the  direction  of  the  ray  within  the 
crystal.  For  this  purpose,  put  «  =  angle  C  O  P  =  azimuth  of  the  principal  section  of  the  plate  reckoned  from 
the  plane  of  primitive  polarization,  0  =  A  P,  ff  =  A  P',  and  let  us  (for  simplicity)  consider  only  at  present  the  case 
when  P  and  P"  are  near,  as  in  nitre,  so  that  arcs  of  circles  may  be  regarded  us  straight  lines,  and  spherical  as  plane 
triangles,  (see  Art.  907.)  Now  if  in  fig.  208  we  put  <f>  for  the  angle  P  N  A,  or  the  angle  made  by  the  plane  of 
ordinary  polarization  with  the  principal  section,we  shall  have  T^=CMA=COP-)-MNO=:COP-)-PNA 

(P  A\2 
^-L  \  x  sin  (PAN  =  £  P  A  P1)2  ;  but  since  N  A  Analyti- 
I  «/  Cally  ex- 

P  A  2  a  0  pressed. 

bisects  the  angle  of  the  triangle  P  A  P'  and  cuts  the  base,  P  N  —  P  P'  X     —  —      —  ,  =  -     —  ;,  and 

P  A  -p  A  P        0  -\-  0' 


4  a°  _  (0  ~ 
(siniPAFr-=i(l-«»PAP')  =  -- 


so  that 


558  LIGHT. 

Light-     A  more  symmetrical  value  of  0  will,  however,  be  had  by  expressing  the  value  of  sin  2  0,  which  being  equal  to  Part  IV. 

"•'V^1*'  4  .  sin  0s  (1  —  sin  0*)  is  immediately  given  by  substitution  of  the  foregoing.     If  we  execute  the  reductions  we  **— -v-» 

shall  find  that,  putting  S  for  ,          -  =  half  the  sum  of  the  sides  of  the  triangle  PAP 


2  (0  -f.  O1)  (0  -  </)       V  S  (S  -  6)  (S  -  0')  (S  -  2  a) 

Sin2f6=      -(^T    -'  -off- 

Now   -  -is  the  well  known  expression  for  the  sine  of  the  angle  PAP'  included 

0  0 

between  the  sides  0,  0',  and  therefore  calling  this  angle  P,  we  have 

(0  +  0')  (0  -  0') 
sm20:=-    _____  -  .smP. 

The  nature  of  this  expression  renders  the  transition  from  plane  to  spherical  triangles  easy,  and  we  may  conclude 
consequently,  that,  in  crystals  where  the  axes  make  any  angle  2  a,  that  if  we  take 

sin  (0  +  0')  .  sin  (0  -  6') 
sin  2  0  =  -  —  -  .  sin  P, 

(sin  2  a)8 

and  ty  =  «  +  0,  we  shall  still  have  the  intensity  of  the  extraordinary  rings  represented  by  A  (sin  2  Y^)2,  and  that 
of  the  ordinary  by  1  —  A  +  A  .  (cos  2  i^)2,  that  is,  1  —  A  (sin  2  ty)*,  their  sum  being,  as  it  ought,  unity. 
1072          The  black  cross  which  divides  the  system  of  the  primary  rings,  is  too  remarkable  a  feature  not  to  require  express 
Form  of  the  notice.     Its  form,  it  is  evident,  must  be  determined  by  the  condition  that  the  line  M  A  shall  be  everywhere  perpendi- 
black  cross  cular  to  C  O  D,  in  which  circumstances  the  locus  of  A  will  be  a  curve  marking  out  its  central  or  blackest  portion. 
in  biaxal      rjne  probiem  then  is  reduced  to  a  purely  geometrical  one,  Required  a  curve  P  A  such  that  a  line  drawn  from  A 
ls  '?"  bisecting  the  angle  between  lines  A  P,  A  P'  drawn  to  two  given  points  P,  P',  shall  always  be  perpendicular  to  a 
given  line  C  O  D.     To  resolve  this,  retaining  the  former  notation,  and  putting   O  M  =  x,  MA.  =  y,   O  A=  r, 
we  have 

x  .  cos  a  -f  y  .  sin  a       N 

cos  A  O  P  =  cos  (A  O  M  -  a)  =  —  -  =  —  , 

r  r 

.  cos  a  —  x  .  sin  re        M 


sin  A  O  P  = 


r  r 


puttino-  N  and  M  for  the  respective  functions  in  the  numerators  of  the  fractions. 
Now  since  PAM  is  half  the  angle  PAP',  it  is  easy  to  see  that  we  must  have  2 


Now  since  PAM  is  half  the  angle  PAP',  it  is  easy  to  see  that  we  must  have  2xangle  O'AM  =  PAO-P'AO. 


-      -  a*  O'^  +  ^ 

But,  cos  P  A  O  =      \re       ;  cos  F  A  O  =    -J 


PO       «M  aM 

and  smPAO  =  sinAOPXp^  =  7^;  smP'AO  =  —  ; 


consequently  we  have,  first, 

«  M    (  S'J  -f-  r'-   -a*       0*  +  r*  —  a5}  a  M 

0^  00'         I  "  2r200 


r«          2r*    (.          < 
and,  secondly, 

cos20AM=^r  =  — 
Now  we  have  further, 


which  substituted  in  the  values  of  sin  2  O  AM  and  cos  2  O  AM  above,  give  the  equations 


(/  -  a?)  .  &  &'  =  r4  +  a*  (M«    -  N!), 

and,  eliminating  9  6'  from  these,  we  obtain 

0»  (y»  _  ^)  .  MN  =  xy  {  r'  +  of  (M8  -  N1)  }. 

In   this  it  only  remains  to  substitute  for  M  and  N  their  values  y  .  cos  a  -  x  .  sin  a,  and  y .  sin  <i  +  t . 
which  done,  the  whole  will  be  found  divisible  by  r\  and  will  reduce  itself  to  the  very  simple  equation 


cos  •, 


L  I  G  H  T.  559 

Light.  c?       .  part  iv. 

x  y  =  a  .  sin  o .  cos  a  =  —  .  sin  2  a. 

The  black  cross  then  is  an  hyperbola,  passing  through  the  poles  P,  P',  and  having  the  planes  of  primitive  pola-  axes  are 
rization,  and  one  perpendicular  to  it  (C  D  and  cd)  for  its  asymptotes,  and  which  as  a  approaches  to  0,  or  90°,  near  it  is  ar 
approaches  nearer  and  nearer  to  its  asymptotes,  with  which  it  at  last  coincides  in  the  limiting  case,  all  which  hyptfboU. 
particulars    are    exactly    conformable  to  fact,  and  may   easily   be  verified  by  turning  a  plate    of  nitre  round 
between   crossed   tourmalines.     When    the  inclination    of  the   axes  is  so  considerable,  that  the  rings  about 
both  poles  cannot  be  seen  at  once,  there  will  arise  modifications  from  the  substitutions  of  the  sines,  &c.  of  arcs 
for  the  arcs  themselves,  which  it  is  not  worth  while  to  enter  into. 

To  return  now  to  the  phenomena  of  dichroism.  That  portion  of  the  light  transmitted  by  a  biaxal  coloured  1073. 
medium  which  has  relation  to  the  optic  axes,  and  which  forms  the  sombre  brushes  of  colour  (in  fig.  203,)  and  Empirical 
the  bright  spaces  which  divide  them,  have  evidently  for  their  analytical  expression  a  function  of  the  form  formula 

Y .  (cos  2  0)2  +  B  .  (sin  2  0)2 ;     (a)  *?$£. 

where  Y  and  B  are  functions  of  X,  and  represent  the  ordinates  of  the  types  of  two  fundamental  tints,  0  represent-  dichroism 
ing  as  before  the  angle  PN  A,  fig.  208,  or  the  angle  made  by  the  plane  of  ordinary  polarization  with  the  principal 
section.  But  besides  this,  the  phenomena  described  by  Dr.  Brewster,  as  exhibited  by  the  iolite,  require  us 
to  admit  two  other  portions,  which  may  be  more  naturally  referred,  not  to  either  of  the  optic  axes  but  to 
the  line  C  O  (fig.  209)  bisecting  them,  and  having  for  its  expression  a  function  of  the  form  a  .  cos  O  A*  + 
b  ..sin  OA2.  In  this  mineral,  when  exposed  to  common  light  (or  to  polarized,  provided  we  place  its  principal  section 
at  right  angles  to  that  of  polarization,)  the  lateral  brushes  A,  B,  fig.  206,  are  blue,  and  the  bright  rays  which 
divide  them,  passing  through  the  poles  P,  P'are  white,  or  yellowish  white,  and  so  far  the  phenomena  agree  with 
the  expression  (a)  if  we  suppose  Y  to  represent  a  bright  yellowish  white,  and  B  a  blue.  But  according  to  that 
expression  alone,  the  blue  spaces  should  be  continued  down  to  the  equator  C  a  b  D,  fig.  206,  and  there  ought 
to  be  two  directions  C  D  and  a  b  in  which  the  mineral  viewed  transversely  to  the  axis  of  the  prism  (which  is 
perpendicular  to  the  plane  Ca6D)  should  appear  yellow,  and  two  others,  m  n  and  p  q,  in  which  it  should 
transmit  a  blue  colour,  while  in  the  direction  of  the  axis  O  it  should  appear  yellow.  Now,  on  the  contrary,  the 
equatorial  colour  is  nearly  uniform  and  pale  yellow,  while  that  along  the  axis  O  is  blue ;  and  in  proceeding 
from  the  equator  toward  the  axis  O  of  the  prism,  the  yellow  diminishes,  and  the  blue  gains  strength,  whether 
we  set  out  from  C  and  D,  or  from  a  and  6,  precisely  as  would  be  indicated  by  the  other  formula 

y  .  (sin  O  A)2  +  6  .  (cos  O  A)2, 

y  representing  a  yellow  white  and  6  a  blue  tint.     If,  therefore,  we  put  O  A  =  v,  the  joint  expression 
T  =  (Y .  cos  2  0"  +  B  .  sin  2  0")  +  (y  .  sin  *3  -f  b  .  cos  **)  ;    (6) 

will  be  found  to  represent  pretty  correctly  the  variations  of  colour  as  far  as  they  can  be  judged  of  by  the  eye. 
Thus,  at  O  where  v  =  o,  and  0  =  90°,  we  have  T  =  Y  -j-  b,  which  may  indicate  either  a  yellow,  a  white,  or  a 
blue,  according  as  we  suppose  Y  or  6  to  be  predominant.  The  fact  being,  that  the  tint  at  O  is  blue,  we  must 
suppose  the  latter  to  express  the  more  decided  colour.  As  we  proceed  from  O  along  the  sections  O  C,  O  D, 
or  O  a,  O  b,  in  both  of  which  sin  2  0  =  o,  we  have 

T  =  (Y  +  y  .  sin  v«)  -j-  6  .  cos  v*  =  (Y  +  b)  +  (y  -  b)  .  sin  »* 

Now  y  expressing  a  yellow  white  and  6  a  strong  blue,  y  —  b  will  express  a  proportionally  vivid  yellow,  and 
therefore  the  blue  tint  Y  +  6  seen  along  the  axis  will  be  diluted  with  more  and  more  yellow  as  we  approach  the 
equator ;  at  P  P',  then,  (by  a  proper  assumption  of  numerical  values)  it  will  be  rendered  nearly  neutral,  after 
which  the  yellow  will  predominate,  and,  at  the  equator,  will  remain  alone  sensible,  the  expression  for  T  then 
becoming  T  =  Y  -j-  y,  at  the  points  C,  a,  b,  D.  Let  us  next  consider  the  case  when  cos  20  =  o,  or  0  =  45°, 
that  is  to  say,  along  the  axes  or  most  intense  lines  of  the  lateral  brushes.  In  this  case  we  have 

T  =  B  +  (y .  sin  v8  4-  6.  cos  v')  =  (B  +  b  .  cos  if)  -f  y  .  sin  v'. 

Now  if  we  suppose  B  and  b  to  represent  blue  tints,  since  (in  the  case  of  iolite)  the  angle  between  the  axes  or 
PP'=62°50'  and  OP  =  31°25',  we  have  in  the  immediate  vicinity  of  the  poles,  (sin  v)4  =  £  nearly,  and 
cos  v1  =  J,  so  that  in  the  immediate  neighbourhood  of  P  the  tint  of  the  most  intense  part  of  the  brushes  will  be 
B  +  |  b  +  J  y,  which,  on  very  reasonable  suppositions  of  the  numerical  values  of  B,  6  and  y  will  denote  a  full 
and  rich  blue.  But  as  we  approach  the  equator  at  m,  n,  p,  q,  cos  vs  diminishing  and  sin  v*  increasing,  the  sombre 
tint  B  is  continually  more  feebly  reinforced  by  the  tint  b  .  cos  v*  and  more  strongly  counteracted  by*y .  sin  v1,  till 
at  length  it  will  be  overpowered,  and  the  colour  in  these  points,  as  in  C,  a,  D,  b,  will  be  yellow  only  somewhat 
less  decided  than  in  the  latter,  its  tint  being  represented  by  T  =  y  +  B  instead  of  y  +  D. 

In  general,  if  we  put  A  for  the  tint  transmitted  along  the  axis  O  of  the  prism,  P  for  that  seen  along  the  poles,      1074 
L  for  that  of  the  lateral  branches  at  their  origin  close  to  the  poles,  and  E  for  the  mean  equatorial  tint,  we  shall  Detemfm*- 
have  for  determining  Y,  y,  B,  b,  the  equations  tion  of  the 

coefficients 

A  =  Y  +  6,    2E  =  2y  +  B  +  Y,  from  lhe 

P  =  Y-f-y.sin<z2  +  b.  cosa*;     L=  B  +y  .  sin  a?+  b  .  cos  a',  observed 

tints* 

on  elimination  from  these,  it  will  appear  that  there  is  an  equation  of  condition  to  be  satisfied,  viz. 

2  (A  -  P)  =  (2  A  -  2  E      P  -f  L)  .  sin  a' ;    (c) 


560  L  I  G  H  T. 

Light.      and  that  supposing  it  satisfied,  one  of  the  tints,  as  y,  will  (so  far  as  these  conditions  are  concerned)  remain     Part  IV. 
-^v-^  arbitrary,  and  the  others  will  be  given  by  the  equation  ^— ^-». 


2b   = 

.     in  which  y  must,  however,  be  such  as  to  render  Y,  B,  b  real  tints,  i.  e.  expressed  by  positive  numbers. 

1075.         To  apply  this,  for  example's  sake,  to  the  case  of  the  iolite,  let  us  regard  every  white  ray  as  consisting  of  two 

Application  complementary  rays  of  bright  yellow  and  bright  blue  of  equal  efficacy ;  and  suppose  that  by  observation  we 

iolit'18          'lav°  asccrtamed  >ts  equatorial  tint  E  to  be  a  very  pale  but  strongly  luminous  yellow  white,  consisting  of  110 

such  yellow  rays,  and  99  such  blue  ones,  producing  a  joint  intensity  =  209.     Moreover,  let  the  tint  seen  along 

the  axis  of  the  prism  (A)  be  a  blue,  of  a  good  colour,  but  considerably  less  intensity,  represented  by  10  such 

yellow  -f-  20  such  blue  rays  =  30.     That  seen  along  the  optic  axes  (P)  to  be  a  white  represented  by  30  yellow 

-f-  36  blue  ==  72,  and  that  of  the  most  intensely  coloured  portions  of  the  lateral  brushes  =  L  to  be  a  stronger 

blue  than  that  seen  in  the  axis  of   the  prism,   such   as  may   be   represented   by  28  yellow  -j-  66  blue  =  94. 

These  numbers  are  chosen  so  as  to  satisfy  the  equation  of  condition,  taking  a  =  30°,  and  if  we  substitute  them 

we  shall  find 

y  +  y  =  114  yellow  +  84  blue  ;       B  -f  y  —  106  yellow  +  1 14  blue  ;       y  -  b  =  104  yellow  +  64  blue, 

y  remaining  indeterminate  ;  if  we  suppose  its  composition  to  be  in  yellow  +  n  blue,  we  may  determine  m  and 
n  by  the  two  conditions  that  b  shall  (as  we  have  before  supposed)  represent,  a  pure  blue  without  any  mixture 
of  yellow,  and  Y  a  very  pale  yellow,  such  as  would  result  from  a  mixture  of  yellow  and  blue  in  the  ratio  of  10 
to  9.  These  conditions  are  satisfied  by  taking  m  =  104  and  n  =  75  ;  so  that  we  have,  finally, 

Y  =     10  yellow  +     9  blue  ;  B  =  2  yellow  -f  39  blue  ; 

y  =  104  yellow  +  75  blue  ;  b  =  0  yellow  +11  blue ; 

and  these  being  taken  for  the  values  of  the  coefficients  in  the  expression  (6)  Art.  1073,  it  will  be  found  on  (rial 
to  reproduce  the  tints  actually  observed.  In  fact,  the  extreme  equatorial  tints  being  y  +  Y  and  y  +  B,  will  be 
respectively  represented  by  114  yellow -j- 84  blue,  and  106  yellow  +  114  blue;  the  former  is  a  very  pale 
yellow,  but  highly  luminous,  being  equivalent  to  30  rays  of  yellow  diluted  with  168  of  white;  while  the  latter  is 
a  blue  so  pale  as  to  be  undistinguishable  from  white,  and  also  highly  luminous,  being  equivalent  to  8  rays  of 
blue  diluted  with  212  of  white. 

1076.  The  reader  will  perceive  that  the  formula  in  question  is  merely  empirical,  and  that  more  numerous  experi- 
Phenomena  ments  than  we  possess  will  be  required  to  establish  or  disprove  it.      It  is  unfortunately,  however,  difficult   to 
exhibited     meet  with  biaxal  crystals  sufficiently  dichromatic  for  the  purposes  of  decisive  experiment,  and  at  the  same  time 
crystal's!118    'arSe   anc^  transparent  enough  to  admit  of  being  cut  into  the  forms  and  examined  in   the  directions  required, 

through  a  thickness  sufficient  for  a  full  developement  of  their  colours.  Such  are  indeed  hardly  less  rare  than 
the  most  precious  gems  ;  and  this  circumstance  is  a  great  obstacle  to  the  advancement  of  our  knowledge  in  one 
of  the  most  interesting  branches  of  optical  inquiry,  which  that  of  dichroism  certainly  deserves  to  be  considered. 
Among  artificial  crystals,  however,  there  is  room  to  suppose  that  subjects  fit  for  such  experiments  may  be  met 
with.  One  remarkable  instance  of  dichroism  among  these  has  been  mentioned  in  the  sub-oxysulphate  of  iron. 
To  this  we  may  add  the  potash-muriate  of  palladium,  which  exhibits  along  the  axis  of  the  four-sided  prism  in 
which  it  crystallizes  a  deep  red,  and  in  a  transverse  direction  a  vivid  green.  (Wollaston,  Phil.  Trans.  1804.  On 
a  new  metal  in  Crude  Platina.)  The  curious  property  of  the  purpurates  of  ammonia,  potash,  &c.  described  by 
Dr.  Prout,  (Phil.  Trans.  1808,)  which  by  transmitted  light  exhibit  an  intense  red,  and  by  reflected,  on  one 
surface,  a  dull  reddish  brown,  and  on  another  a  splendid  green,  appears  referable,  not  so  much  to  the  principles 
<>f  dichroism  properly  so  called,  as  to  some  peculiar  conformation  of  the  green  surfaces,  producing  what  may  be 
best  termed  a  superficial  colour,  or  one  analogous  to  the  colour  of  thin  plates,  and  striated  or  dotted  surfaces. 
A  remarkable  example  of  such  superficial  colour,  differing  from  the  transmitted  tints,  is  met  with  in  the  green 
fluor  of  Alston-moor,  which  on  its  surfaces,  whether  natural  or  artificial,  exhibits,  in  certain  lights,  a  deep  blue 
tint,  not  to  be  removed  by  any  polishing. 

1077.  Dr.  Brewster  has  shown  that  the  action  of  heat  often  modifies  in   a  very  remarkable   manner  the  colour  of 
Unequal       doubly  refracting  crystals,  producing  a  permanent  change  in  the  scale  of  absorption  of  the  crystals  as   aflbcting 
effects  of     one  Qf  tne  penej]s  and  not  the  other.     Thus,  having  selected  several  crystals  of  Brazilian  topaz  which  displayed 
colour"  of*  no  change  of  colour  by  exposure   to  polarized  light,  (and  in   which,  of  course,  the  types  of  both  absorptions 
the  two        must  have  been  alike,)  and  bringing  them  to  a  red  heat,  or  even  boiling  them  in  olive  oil,  or  mercury,  they  expe- 
pencils.         rienced  a  permanent  change,  and  had  acquired  the  property  of  absorbing  polarized  light  unequally.     He  then 

took  a  topaz  in  which  one  of  the  pencils  was  yellow  and  the  other  pink  ;  and  by  exposing  it  to  a  red  heat,  he 
found  the  extraordinary  pencils  more  powerfully  acted  on  than  the  ordinary,  the  yellow  colour  being  discharged 
entirely  from  the  one,  while  only  a  slight  change  was  produced  in  the  pink  tint  of  the  other.  This  change  of  colour 
in  the  topaz  by  heat  (though  not  its  intimate  nature)  is  well  known  to  jewellers,  who  are  in  the  habit  of  thus 
developing  in  this  gem  a  colour  more  highly  prized.  It  is  remarkable,  that  while  hot  the  topaz  is  perfectly  colour- 
less, and  acquires  the  pink  colour  gradually  in  cooling.  By  the  repeated  action  of  very  intense  heat  Dr. 
Brewster  was  never  able  to  modify  or  remove  this  permanent  pink  tint.  How  far  violent  compression,  slow 
application,  and  abstraction  of  the  heat,  or  other  modifying  circumstances,  might  prevent  its  developement,  i'. 


LIGHT.  5G1 

Light,     would  be  interesting  to  examine  ;    since  we  cannot  help  being  otherwise  struck  by  the  force  of  the  argument    Part  IV. 
«-Y~~^  geologists  may  draw,  from  the  existence  in  rocks  of  a  mineral  which  mere  elevation  of  temperature  unaccompanied  '— v— •* 
with  change  of  composition,  thus  irrevocably  alters. 

One   general  character  of  all  dichroite  bodies  is,  that  when  natural  light  is  transmitted  through  a  plate  of     1078. 
sufficient  thickness,  in  any  direction  not  coincident  with  one  of  the  optic  axes,  the  emergent  beam  is  wholly  or  General 
partially  polarized  by  reason  of  the  unequal  action  of  the  medium  on  the  two  pencils,  and  the  consequent  sup-  jj1*^-,. 
pression  of  one  of  them.     And,  in  general,  whatever  cause  tends   to  interfere  unequally  with  their  free  trans-  crysla|s. 
mission  through  a  medium,  will  produce  a  similar  effect.     Thus,   for  example,  if  the  continuity  of  a  doubly  Effects  of 
refracting  medium  be  interrupted  by  a  film  of  any  uncrystallized  substance,  since  the  two  pencils  by  reason  of  ancrystal- 
their  angular  separation  are  incident  on  this  film  at  different  angles  ;  and  since,  moreover,  their  relative  refractive  J.™^"'1 
indices,  with  respect  to  the  medium  composing  the  film,  differ,  they  will  undergo  partial  reflexion  at  the  film  in  ^s  " 
different  proportions,  and  thus  an  inequality  will  arise  in  the  parts  transmitted.     If  the  refractive  index  of  the 
film  be  precisely  equal  to  the  ordinary  refractive  index  of  the  crystal  (supposed,  for  simplicity,  to  be  uniaxal) 
the  ordinary  ray,  it  is  evident,  will  undergo  no  disturbance   or  diminution,  while  the  extraordinary  will    be 
changed  in  direction  and  diminished  in  intensity  by  partial  reflexion  at  its  ingress  and  egress,  at  every  such  film 
which  may  exist  in  the  medium.     If  the  films  be  extremely  numerous,  and  if,  moreover,  they  be  not  disposed 
in  planes,  but  in  undulatory  or  irregular  surfaces  through  the  medium,  this  will  make  no  difference,  so  far  as  the 
ordinary  ray  is  concerned,  which  will  still  pass  undisturbed  through  the  system,  (except  so  far  as  any  opacity  in 
the  matter  of  the  films  may  extinguish  a  portion  of  it ;)  but  the  extraordinary  ray  will  be  rendered  confused, 
and  dispersed,  its  egress  from  the  films  not  being  performed  (by  reason  of  their  curvature)  at  the  same  angles 
as  its  ingress,  and  that  irregularly,  according  to  their  varying  inclination.     Hence  will  arise  a  phenomenon  pre-  Phenomena 
cisely  such  as  is  presented  by  the  agate,  and  other  irregularly  laminated  bodies,  through  plates  of  which,  if  a  °  a=att 
luminary  be  viewed,  it  is  seen  distinctly,  but   as  if  projected  on   a  curtain   of  nebulous   light  ;    and    if  ex- 
amined with  a  tourmaline,  or  doubly  refracting  prism,  the  distinct  image,  and  the  nebulous  light,  are  found  to 
be  oppositely  polarized.     If  we  examine  a  piece  of  agate  with  a  magnifier,  the  laminated  structure  and  unequal 
refraction  of  the  laminae  are  very  apparent ;    it  appears  wholly  composed  of  a  set  of  exceedingly  close  layers, 
not  arranged  in  planes,  but  in  undulating  or  crinkled  lines  like  a  number  of  figures   of  333333   placed  close 
together.     The  planes  of  polarization  of  the  nebulous  and  distinct  image  are  parallel  and  perpendicular  to 
the  general  direction  of  the  layers,  which  through  any  very  small  portion  of  the  substance  is  generally  pretty 
uniform. 

But  the  film  interposed  may,  itself,  be  crystallized,  and  inserted  between  adjacent  portions  of  a  regular  crystal,  1079. 
according  to  the  crystallographic  laws  which  regulate  the  juxtaposition  of  the  molecules  at  the  common  surfaces  Action  of  a 
of  macled  or  hemitrope  crystals.  Let  A  D  E  F  (fig.  210)  be  such  a  plate  interrupted  by  a  crystallized  lamina  frysta"'z 
B  C  E  F,  bounded  by  parallel  planes,  and  let  us  consider  what  will  happen  to  a  ray  S  a  incident  at  a.  It  is  "^ 
evident,  that  were  the  crystallized  lamina  away,  or  were  its  molecules  homologously  situated  with  those  of  the  pjg.'  210. 
portions  on  either  side  of  it ;  in  the  latter  case,  we  should  have  an  uninterrupted  crystal ;  in  the  former,  two 
prisms  disposed  with  their  principal  sections  parallel,  and  acting  in  opposition  to  each  other;  in  either  case,  the 
emergent  ordinary  and  extraordinary  pencils  separated  by  double  refraction  at  the  first  surface  will  emerge 
parallel  to  the  incident  ray,  and  therefore  to  each  other.  But  the  principal  section  of  the  crystallized  film  being 
non-coincident  with  those  of  the  two  prisms  ABE,  CFG,  it  will  alter  the  polarization  of  the  portions  ab,  ac ; 
and  in  place  of  their  being,  as  in  the  former  case,  each  refracted  singly  by  the  second  prism  CFG,  they  will  now 
each  be  refracted  doubly,  so  that  in  place  of  two  emergent  rays  there  will  now  be  four.  The  subdivision  of  the 
rays  within  the  interposed  lamina  may  evidently  be  disregarded,  for  they  will  be  refracted  in  passing  from  the 
film  into  the  second  prism  in  the  same  direction,  where  contiguous,  as  they  would  were  an  infinitely  thin  plate  of 
air  interposed.  Now,  in  that  case,  they  would  emerge  from  the  film  in  pairs  respectively  parallel  to  the  incident 
rays  a b,  ac,  and  therefore  to  each  other.  Hence  the  refraction  at  the  second  prism  will  be  precisely  the 
same  as  if  the  lamina  were  suppressed,  and  in  its  place  the  rays  ab,  ac  had  received  at  a  the  polarizations 
they  acquire  by  its  action.  Now,  these  being  in  opposite  planes,  it  is  evident  that  each  of  the  rays  a  b,  a  c 
would  undergo  both  an  ordinary  and  an  extraordinary  refraction.  Let  us  denote  these  four  emergent  pencils 
so  arising  by  O  O,  O  E,  E  O,  E  E,  and  suppose  a  b  to  be  the  direction  taken  by  the  ordinary  refracted  portion 
of  S  a,  and  a  c  that  of  the  extraordinary.  Then,  since  O  O  has  been  refracted  ordinarily  by  the  prism  CFG, 
and  was  incident  on  it  in  the  direction  of  the  ordinary  ray  a  b,  its  direction  on  emerging  will  be  parallel  to  S  a. 
Similarly,  E  E  is  refracted  extraordinarily,  and  being  incident  in  the  direction  6  c  of  the  extraordinary  portion 
of  S  a,  it  also  will  emerge  parallel  to  S  a,  and  thus  the  two  rays  O  O,  E  E  will  emerge  parallel,  and  their 
systems  of  waves  will  be  superposed.  But  the  portions  O  E  and  E  O,  the  one  being  incident  in  the  ordinary 
direction,  but  refracted  extraordinarily,  the  other  incident  in  the  extraordinary  direction  and  refracted  ordinarily, 
will  neither  emerge  parallel  to  the  original  ray  S  a,  nor  to  each  other ;  and  this  will  give  rise  to  two  lateral 
images,  one  on  each  side  of  the  central  or  direct  image,  which  will  have,  moreover,  an  intensity  equal  (except 
in  extreme  cases)  to  the  sum  of  those  of  the  lateral  images. 

If  the  film  E  B  C  F  be  very  thin,  or  if  either  of  its  optic  axes  be  nearly  coincident  with  the  direction  in  which      1080, 
the  light  traverses  it,  the  difference  of  paths  and  velocities  within  it  will  give  rise  to  an  interference  of  the  pairs  Phenomena 
of  rays  going  to  form  either  pencil  emergent  from  the  film,  and  thus  will  arise  the  colours  of  the  rings  in  each  ot  'nter- 
image.     Those  on  either  side  the  central  one  will  be  consequently  tinged  with  the  respective  colours  of   the  ^j^j 
primary  and  complementary  set  of  rings  ;  while  the  central  image,  being  formed  by  the  precise  superposition  of  Spar 
two  similar  complementary  pencils  will  appear  white. 

All  these  phenomena  actually  occur,  and  have  been  described  by  Dr.  Brewster,  and  explained  by  him  on  the 
principles  here  laid  down,  in  certain  not  uncommon  specimens  of  Iceland  spar,  which  are  interrupted  by  such 

VOL   iv.  4  D 


562  LIGHT. 

Light,     hemitrope  films,  passing  through  the  longer  diagonals  of  opposite  faces  of  the  primitive  rhomb.     If  we  look  at 
•— — Y— **  a  candle  through  such  an  interrupted  rhomb,  it  will  be  seen  accompanied  by  a  pair  of  lateral  images  such  as 
here  described,  and  exhibiting  frequently  the  complementary  tints  with  great  splendour. 

1081.  If  tne  luminary  from  which  the  ray  S  a  issues  be  small,  the  lateral  images  will  be  separated  by  a  dark  interval 
Phenomena  from  each  other  and  from  the  central  one,  but  if  large  they  will  overlap.     If  infinite  (as  where  the  uniform  light 
of  idio-        of  the  sky  is  viewed)  all  the  images  will  be  superposed.     But  the  field  of  view  will   not  necessarily  be  uniform 
cyclopha-     an(j  w)jjte.    The  central  image  will  form  an  intense  white  screen,  or  ground,  on  which  will  be  projected  the  lateral 
crystals        ones.     Now,  if  the  film   be  so  constituted  as  to  have  within  the  visible  field  of  view  of  one  only  of  the  lateral 

images  the  pole  of  one  of  its  sets  of  rings,  (which  will  be  the  case  whenever  one  of  its  optic  axes  is  not  very 
remote  from  perpendicularity  to  the  surface  of  the  plate  A  D,  so  as  to  admit  of  one  of  the  rays  O  E  or  E  O 
traversing  the  film  in  the  direction  of  its  axis,)  that  set  of  rings  will  not  be  seen  projected  centrally  on  the  cor- 
responding set  complementary  to  it  of  the  other  lateral  image,  by  reason  of  the  angular  separation  of  these  two 
images.  Of  course  its  colours  will  not  be  neutralized,  and  it  will  be  visible  per  se,  though  very  faint,  being 
diluted  by  the  whole  white  light  of  the  central  image  (O  O,  E  E)  and  by  the  whole  visible  and  nearly  uniform 
portion  of  the  other  lateral  one  (O  E.) 

1082.  This  is  not  the  only  way  in  which  a  crystal  perfectly  colourless  may  exhibit  its  sets  of  rings  by  exposure  to 
Fig.  211.      common  daylight  without  previous  polarization,  or  without  subsequent  analysis  of  the  transmitted  pencil.     The 

general  mass  of  the  crystallized  plate  may  have  one  of  its  optic  axes  in  the  direction  of  the  visual  ray,  as  in 
fig.  211,  and  the  portion  of  it  C  Drfc  included  between  two  films  ttCcb  and  DdeE  will  then  form  precisely 
such  a  combination  as  that  above  described,  and  will  exhibit  a  set  of  rings  feeble  in  proportion  to  the  rarity  and 
minuteness  of  the  films,  and  the  consequently  small  area  of  their  outeropping  surfaces  B  C,  D  E.  These  are  not 
hypothetical  cases.  Dr.  Brewster  states  himself  to  have  met  with  specimens  of  nitre  exhibiting  their  rings  per 
SK.  Such  are  rare.  But  in  the  bicarbonate  of  potash  it  is  an  accident  of  continual  occurrence ;  and,  indeed, 
almost  universal.  The  films  in  both  cases  are  easily  recognised,  and  their  position  and  that  of  the  system  of 
rings  seen  leave  no  doubt  of  the  correctness  of  the  explanation  here  given.  Such  crystals,  of  which  more  will 
no  doubt  be  hereafter  recognised,  may  be  termed  idiocyclophanous  till  a  better  term  can  be  thought  of. 


§  XII.     On  the  effects  of  Heat  and  Mechanical  Violence  in  modifying  the  action  of  Media  on  Light,  and  on  the 
application  of  the  Undulatory  Theory  to  their  explanation. 

1083.  It  was  ascertained  independently,  ami  about  the  same  time  by  Dr.  Seebeek  and  Dr.  Brewster,  that  when  glass, 
General        which  in  its  ordinary  state  offers  none  of  the  phenomena  of  double   refracting   media,   is   heated   or  cooled 
account  of    unequally,  it  loses  this  character  of  indifference,  and  presents  phenomena  of  coloration,  &c.  analogous,  in  many 

he  phe-  respects,  to  those  exhibited  by  doubly  refracting  crystals.  If  the  heat  communicated  be  below  the  temperature 
!na'  at  which  glass  softens,  the  effect  is  transient,  and  vanishes  when  the  glass  attains  a  uniform  temperature 
throughout  its  substance,  whether  by  the  equable  distribution  of  the  caloric  throughout  its  mass,  or  by  its 
abstraction  in  cooling.  But  if  the  temperature  communicated  be  so  high  as  to  allow  the  molecules  of  the  glass 
to  yield  to  the  mechanical  forces  of  dilatation  and  contraction  produced  in  the  act  of  cooling  and  take  a  new 
arrangement,  the  effect  is  permanent,  and  glass  plates  so  prepared  have  many  points  of  resemblance  with  crys- 
lallixed  bodies.  Dr.  Brewster  afterwards  ascertained,  that  mechanical  compression  or  dilatation  applied  to 
glass,  jellies,  gums,  and  singly  refractive  crystals  (such  as  fluor  spar,  &c.)  is  capable  of  imparting  to  them  the 
same  characters.  If  the  medium  to  which  the  pressure  is  applied  be  perfectly  elastic,  like  glass,  the  effect,  like 
that  of  heat,  is  transient.  But  if  during  the  continuance  of  the  compression  or  dilatation,  the  particles  of  the 
medium  are  allowed  to  take  their  own  arrangement  and  state  of  equilibrium,  then  when  the  external  force  is 
withdrawn  a  permanent  polarizing  character  will  be  found  to  exist. 

1084.  As  periodical  colours  are  not  produced  in  phenomena  of  this  class  without  a  resolution  of  the  incident  light 
Accompa.     into  two  pencils  moving  with  different  velocities,  and  as  a  difference  of  velocities  is  invariably  accompanied  with 

*{  by  a  difference  of  refraction  at  inclined  surfaces,  it  might  be  expected  that  media  thus  under  the  influence  of  heat 

fraction"1"  or  Pressure  should  become  doubly  refractive.     This  has  been  verified  by  direct  experiment  by  M.  Fresnel,  who 
has  shown  that  a  peculiar  species  of  double  refraction  is  thus  produced. 

1085.  As  the  unusual  heating  or  cooling  of  glass   and   other   substances,  is   well  known  to  produce    in  the  parts 
Effect  of  heated  or  cooled  a  corresponding  inequality  of  bulk,  and  thus  to  bring  the  parts  adjacent  into  a  state  of  strain  in 
heat  ana-  an  respects  analogous  to  that  arising  from  mechanical  violence,  and  as,  in  fact,  the  effects  of  heat  in  communi- 
tha^of10  cating  double   refraction  to   glass,  whether  transient  or  permanent,  are  all,  as  we  shall  see,   (with  one   very 
pressure  obscure  and  doubtful  exception)  commensurate  with  the  amount  of  the   strain  thus  transiently  or  permanently 

induced,  we  have  little  hesitation  in  regarding  the  inequality  of  temperature  as  merely  the  remote,  and  the 
mechanical  tension  or  condensation  of  the  medium  as  the  proximate  cause  of  the  phenomena  in  question,  and 
are  very  little  disposed  to  call  in  the  agency  of  a  peculiar  crystallizing  fluid,  endowed  with  properties  analogous  to 
those  of  magnetism,  electricity,  &c.,  to  account  for  the  phenomena,  still  less  to  regard  media  under  the  influence  of 
heat  or  pressure  as  in  any  way  thereby  rendered  more  crystalline  than  in  their  natural  state  of  equilibrium. 

1086.  In  gasiform,  or  fluid  media,  no  such  phenomena  are  observed  to  be  developed  by  either  heat  or  pressure  ;  the 
reason  is  obvious,  the  pressure  is  equally  distributed  in  all  directions,  and  the  elasticity  of  the  ether  (on  the 
undulatory  hypothesis)  preserves  its  uniformity. 

But  in  solids  the  case  is  different.     The  molecules  cannot   shift  their  places  one  among  the  other,  and  the 


,.   on 


LIGHT.  563 

effect  of  a  compression  in  any  direction  is,  first,  to  urge  contiguous  particles  nearer  together  in  that  direction,     I>art  IV" 
'  and  thereby  to  call  into  action  their  repulsive  forces,  more  than  in  the  natural  state,  to  maintain  the  equilibrium;  '"••"V"" 
secondly,  but  much  more  slightly  to  urge  contiguous  particles  in  a  direction  perpendicular  to  that  of  the  pressure  M°de  °f 
laterally  asunder,  by  reason  of  the  increase  of  the  oblique  repulsive  force  developed  by  the  approach  of  the  mole-  ^,5°",,.°  o 
cules  in  the  line  of  pressure  to  those  which  lie  obliquely  to  that  line.     But  this  action,  which  in  fluids  would  the  mole- 
cause  a  motion  of  the  lateral  particles  out  of  the  way,  in  solids  is  ultimately  equilibrated  by  an   increase  of  the  cules  of 
attractive  forces  of  the  adjacent  molecules  in  a  line  perpendicular  to  the  line  of  pressure  ;   and  thus  we  see  that  solR's- 
every  external  force  applied  to  a  solid  is  accompanied  with  a  condensation  of  its  particles  in  the  direction  of  the 
force  and  a  dilatation  in  a  perpendicular  direction.     It  is  probable,  however,  that  this  latter  is  extremely  minute, 
on  account  of  the  rapid  diminution  of  the  molecular  forces  by  increase  of  distance,  rendering  the  diagonal  action 
insensible.     But  the  former  may  easily  be  conceived  to  produce  in  the  ether,  in  virtue  of  its  connection  (what- 
ever it  be)  with  the  molecules  of  refracting  media,  a  difference  of  elasticity  in  the  two  directions  in   question, 
accompanied  with  all   the  necessary  concomitants  of  interfering  pencils,   periodical  colours,  and  double  refrac- 
tion     The  effect  of  dilatation  will  be  the  converse  of  that  of  compression,  the  direction  of  maximum  elasticity 
in  the  one  case  being  that  of  minimum  in  the  other. 

These  views  are  in  perfect  accordance  with  the  experiments  described  by  Brewster  and  Fresnel  on  compressed      1087 
and  dilated  glass.     According  to  the  former  (Phil.  Trans.   1816.  vol.  106)  the  effect  of  pressure  on  the  opposite  Effects  of 
edges  of  a  parallelepiped  of  glass  is  to  develope  in  it  "  neutral"  and  "  depolarizing  axes,"  the  former  parallel  compression 
and  perpendicular  to  the  direction  of  the  pressure,  the  latter  45°  inclined  to  them  ;  in  other  words,  a  parallelepiped  described. 
of  glass  so  compressed,  will  when  exposed  to  a  ray  polarized  in  the  plane  parallel  or  perpendicular  to  the  sides 
to  which  the  pressure  is  applied,  produce  no  change  in  its  polarization  and  develope  no  periodical  colours,  while 
if  polarized  in  45°  of  azimuth  with  respect  to  those  sides,  it  will  develope  a  tint,  descending  in  the  scale  of  the 
coloured  rings  as  the  pressure  increases. 

In  this  case,  if  the  pressure  be  uniformly  applied  over  the  whole  length  of  each  opposite  side,  the  elasticity  of  the      1088. 
ether  in  every  point  of  the  plate  will  be  uniform  in  either  direction  at  every  point  of  the  plate,  being  a  maximum  in  Explanation 
one,  and  a  minimum  in  that  at  right  angles  to  it.   The  incident  light  therefore  if  polarized  in  azimuth  a°  will  resolve  ™,  l?e  un~ 
itself  into  two  pencils  of  unequal  intensity  (viz.  cos  a4  and  sin  a*)  polarized  in  these  two  planes,  and  differing  at  doctrine. 
their  egress  by  an  interval  of  retardation  proportional   to  t  x  (y1  —  v),  where  t  is  the  thickness  traversed,  and 
t>'  —  v  the  difference  of  velocities  of  the  pencils,  which  when  received  on  a  double  refracting  prism  will  (as  in  the 
case  of  a  crystallized  plate  (Art.  969)  give  rise  to  complementary  periodical  tints  in  the  two  images,  the  extra- 
ordinary image  vanishing  when  a  =  0,   or  90,  and  the   contrast   being   a  maximum   at  45°.     It  is,  of  course, 
extremely  difficult  to  give  such  a  perfect  equality  of  pressure,  so  that  we  must  not  be  surprised   if  a  perfect 
uniformity  of  tint  over  the  whole  surface  of  the  glass  should  not  take  place.     In  the   experiment,  however, 
described  by  Dr.  Brewster  (Prop.  I.  of  the  Memoir  cited)  this  seems  to  have  been  the  case. 

If  we  suppose  the  elasticity  of  the  ether  in  compressed  glass  less  in  the  direction  of  the  force  applied  (and      1089. 
where  consequently  the  medium  is  densest,  according  to  the  general  law)  than  in  the  perpendicular,  the  contrary 
will  be  the  case  in  dilated.     Hence,  supposing  the  forces  equal,  in  two  similar  plates,  the  extraordinary  waves,  or 
those  whose  vibrations  are  performed  in  the  direction  of  the  pressure,  and  which  are  therefore  polarized  at  riffht 
angles  to  that  direction,  will  advance  most  rapidly  in  the  former  case,  the  ordinary  in  the  latter.     Consequently,  if  Opposite 
we  regard  the  interval  of  retardation  or  the  tint,  t  ((/  —  u)  as  negative  in  the  former  case,  it  will  be  positive  in  the  effects  of 
latter  ;  and  the  tints  in  the  two  cases  will  present  the  opposite  characters  of  those  exhibited  by  doubly  refracting  compression 
crystals  of  the  two  classes  described  in  Art.  940,  et  seq.  see  also  Art.  803,  as  negative  and  positive,  or  repulsive  a.n<*  t"lata" 
and  attractive.     Two  such  plates,  therefore,  placed  homologously,  or  with  the  directions  of  the  forces  coincident,    " 
ought  to  neutralize  each  other,  and  if  crossed  at  right  angles  should  reinforce  each  other  ;  and  in  general,  if  t  be 
the  thickness  and  f  the  compressing  force  applied  to  any  plate  (supposing  the  difference  of  velocities  to  be  pro- 
portional to  the  force,  and  regarding  dilating  forces  as  negative)  we  shall  have  for  homologously  situated  plates 

T  =  tint  polarized  by  any  number  of  plates 

=  (f.t+f.t'+f".t"  +  &C.)  per^olion. 

In  the  case  of  crossed  plates  the  thicknesses  of  those  placed  transversely  are  to  be  regarded  as  negative,  just  as  in 
the  case  of  the  superposition  of  crystallized  plates.  All  these  results  are  conformable  to  the  experiments 
of  Dr.  Brewster. 

The  phenomena  of  contracted  and  dilated  glass   may  most  easily  and  conveniently  be  produced  by  bending      1090. 
a  long  parallel  plate  of  glass  having  its  longer  edges  polished,  and  passing  the  light  through  them  across  its  Tints  pro- 
breadth.     In  this  case,  as  in  all  cases  of  flexure,  the  convex  surface  is  in  a  state  of  dilatation,  and  the  concave  of  duce<'  ''.v 
compression,  while  there  exists  a  certain  intermediate  line  or  boundary  between  these  oppositely  affected  regions    ianss"fl(ac 
in  which  the  substance  is  in  its  natural  state  of  equilibrium,  and  on  both  sides  of  which  neutral  line  the  degree 
of  strain  increases  as  we  recede  from  it  towards  either  surface.     Fig.  212  is  a  section  of  such  a  bent  plate,  Fig.  212. 
much  exaggerated,  through  which  light,  polarized  in  a  plane  45s  inclined  to  its  length,  has  been   passed   and 
analyzed  as  usual.    The  neutral  line  is  marked  by  a  divided  black  stripe,  and  the  tints  on  either  side  of  it  descend 
in  Newton's  scale,  being  arranged  in  stripes  disposed  according  to   the  lines  11,  22,33,  44,  &c.     The   tints, 
however,  on  opposite  sides  of  the  neutral  line  have  opposite  colours,  being  positive  on  the  side  of  the  dilatation, 
or  towards  the  convexity,  and  negative  on  the  compressed  or  concave  side.     In  a  plate  of  glass  1.5  inch  broad,  stale  of 
0.28  thick  and  six  inches  long,  Dr.  Brewster  developed  seven  orders  of  colours  before  the  glass  broke  with  the  strain  ascer- 
bending  force  applied.     This  experiment  affords  an  exceedingly  beautiful  illustration  of  the  action  of  compressing  taine(l  ty 
and  bending  forces  on  solids,  and  furnishes  ocular  evidence  of  the  state  of  strain  into  which  their  several  parts  tlle  tints 


564 


LIGHT. 


Light. 


1091. 

Effects  of 
several  co- 
existing 
strains. 

1092. 
Pressure 
applied  at  a 
point. 


1093. 
Effects  of 
vibration. 


1094. 

Polarization 
by  com- 
pressed 
jellies 


1095. 

Transient 
effects  of 
heat  below 
the  soften- 
ing point. 


1096. 
Case  of  a 
rectangular 
plate  of 
glass  heated 
at  one  edge. 


Fig.  213. 

1097. 
\ction  of 
heat  in 

straining 
the  glass. 


are  brought  by  external  violence.    The  ingenuity  of  Dr.  Brewster  has  not  overlooked  its  application  to  the  useful    Part  IV. 
and  important  object  of  ascertaining  the  state  of  strain  and  pressure  on  the  different  parts  of  architectural  struc-  — •^^-*— 
lures,  as  stone  bridges,  timber  framings,  &c.,  by  the  use  of  glass  models  actually  put  together  as  the  buildings 
themselves.     We  must  recollect  always,  however,  that  the  information  thus  afforded  will  only  be   distinct  when 
the  load  intended  to  be  sustained  is  many  times  the  weight  of  the  materials. 

If  a  plate  of  glass  be  subjected  to  several  distinct  compressions  and  dilatations  in  different  directions,  Dr. 
Brewster  finds,  that  its  action  will  be  the  same  as  the  combined  action  of  several  plates  each  subjected  to  one  of  the 
forces  employed.  Thus  a  square  of  glass  compressed  equally  on  all  its  four  edges  exerts  no  polarizing  action. 

If  a  pressure  be  applied  at  a  single  point  of  a  mass  of  glass,  or  rather  at  two  opposite  points,  it  will  diverge 
from  these  points  in  all  directions  into  the  mass,  and  the  lines  of  equal  pressure,  which  are  in  fact  the  isochro- 
matic  lines,  must  have  their  form  determined  in  some  measure  by  the  figure  of  the  compressing  screw  or  tool  at 
its  point  of  contact  with  the  glass,  for  this  figure  regulates  the  form  and  curvature  of  the  indentation  immediately 
under  it.  Dr.  Brewster  has  figured  several  of  the  curves  produced  by  the  application  of  such  pressure  to  dif- 
ferent parts  of  the  same  parallelepiped  of  glass,  for  which  the  reader  is  referred  to  his  Paper,  as  well  as  for  a 
variety  of  beautiful  figures  produced  by  crossing  plates  differently  strained. 

M.  Biot  has  observed,  that  in  some  instances  glass  maintained  in  a  state  of  vibration  by  the  action  of  a  bow 
or  otherwise,  depolarizes  light,  i.  e.  restores  the  vanished  pencil.  This  is  a  necessary  consequence  of  the  alter- 
nate compressions  and  dilatations  which  follow  each  other  in  rapid  succession  in  all  the  vibrating  molecules. 
Nodal  lines  (see  ACOUSTICS)  being  exempt  from  such  variations  of  density  ought  to  be  marked  by  black  bands, 
and  may  thus,  perhaps,  be  rendered  evident  to  the  eye. 

When  masses  of  jelly  (especially  of  isinglass)  are  pressed  between  plates  they  acquire  a  polarizing  action.  If 
dilated  by  proper  management,  and  in  that  state  allowed  to  dry  and  harden,  the  character  so  impressed, 
according  to  Dr.  Brewster,  is  permanent  when  the  dilating  force  is  removed ;  to  explain  which,  we  must  consider 
that  the  exterior  coats  indurate  more  rapidly  than  the  interior,  and  when  they  have  acquired  the  con- 
sistency of  a  solid,  they  will  be  capable  of  resisting  the  subsequent  contraction  of  the  interior  portions  and  keep- 
ing them  in  a  dilated  state,  even  when  the  original  dilating  force  is  removed.  That  force  only  served  to  deter- 
mine the  figure  and  dimensions  of  the  exterior  crust,  and  when  once  that  crust  is  fully  formed  and  indurated,  it 
becomes  capable  of  maintaining  them  without  the  further  aid  of  the  cause  which  gave  them  rise.  The  polarizing 
power  of  isinglass  thus  developed  is  very  great,  and  even  exceeds  that  of  some  doubly  refractive  crystals,  such  as 
beryl ;  a  plate  of  isinglass  whose  thickness  is  624  polarizing  the  tint  which  would  be  reflected  by  a  plate  of  air 
whose  thickness  is  unity,  while  a  plate  of  beryl  parallel  to  the  axis,  to  polarize  the  same  tint,  will  require  a 
thickness  =  720.  Glass  compressed,  or  dilated,  by  an  equal  force,  would  require  a  thickness  (according  to 
Dr.  Brewster)  =  12580  to  produce  the  same  tint. 

We  come  now  to  consider  the  transient  effects  of  unequal  temperature  below  the  softening  point  of  glass. 
The  immediate  effect  of  an  increase  or  diminution  of  temperature  in  one  point  of  a  piece  of  glass,  is  to  produce 
a  mechanical  strain  on  all  the  surrounding  part,  which  if  the  difference  of  temperature  is  considerable,  is  of  the 
utmost  violence,  and  capable  of  breaking  asunder  the  thickest  pieces  of  glass ;  an  effect  with  which  every  one  is 
familiar.  Now,  as  we  know  that  strain  alone  developes  a  polarizing  action,  the  rule  of  philosophy,  "  non  phires 
causas  admitti  debere,"  Sfc.  which  forbids  the  admission  of  a  second  cause  when  one  adequate  to  the  effect  is 
known  to  be  in  action,  will  hardly  justify  us  in  attributing  a  peculiar  action  to  the  caloric,  independent  of  its 
power  of  altering  the  dimensions  of  matter. 

When  a  heated  iron  bar  is  applied  along  the  edge  of  a  parallelepiped  of  glass  held  in  a  polarized  beam, 
analyzed  as  usual,  the  vanished  image  is  restored  in  various  degrees  of  intensity  in  different  parts  of  the  glass. 
The  neutral  axes  are  parallel  and  perpendicular  to  the  heated  edge,  and  the  axes  in  whose  azimuth  the  tint 
polarized  is  the  strongest,  at  4!>°  of  inclination.  If  held  in  that  azimuth,  the  first  effect  of  the  heat  is  to  produce 
a  line,  or,  as  it  were,  a  wave  of  white  light  at  the  heated  edge,  which  advances  gradually  upon  the  glass,  driving 
before  it  a  dark  and  undefined  wave.  Nearly  at  the  same  instant,  and  long  before  the  slightest  increase  of  tem- 
perature can  have  reached  the  further  extremity  of  the  glass  plate,  a  similar  but  fainter  white  wave  advances  from 
the  edge  opposite  to  the  heated  one,  driving  before  it  a  similar  undefined  dark  wave  ;  and  at  no  perceptible 
interval  of  time  another  white  fringe  appears  in  a  very  diluted  state  about  the  centre  "of  the  plate,  advancing 
equally  towards  the  heated  edge  on  one  side  and  that  most  remote  on  the  other,  and  thus  condensing  the  two 
undefined  dark  waves  into  two  black  fringes.  The  white  tints  are  succeeded  by  tints  of  a  lower  order  in  the 
scale  of  colour,  yellow,  red,  purple,  blue,  &c.,  till  at  length  the  whole  scale  of  the  colours  of  thin  plates  is  seen 
arranged  in  four  sets  of  fringes  parallel  to  the  heated  edge,  and  having  for  their  origins  the  black  fringes  above 
mentioned.  At  the  same  time,  other  lateral  fringes  are  produced  along  the  edge  perpendicular  to  the  heated  one. 
Thus  in  all  six  sets  are  seen  ;  two  exterior,  viz.  those  parallel  to  the  heated  edge,  and  outside  of  the  black  fringes; 
two  interior,  in  the  same  direction,  but  between  the  black  fringes  ;  and  two  terminal,  along  the  lateral  edges. 
The  whole  phenomena  is  as  represented  in  fig.  213.  The  fringes  along  the  heated  edge  AB  are  most  distinct 
and  numerous,  those  along  the  opposite,  C  D,  less  so(  and  the  interior  and  terminal  fringes  least  of  all. 

As  glass  is  an  extremely  bad  conductor  of  heat,  and  as  culinary  heat  is  propagated  through  glass  entirely  by 
conduction,  it  follows,  that  the  sudden  application  of  an  elevated  temperature  to  the  edge  A  B  must  produce  a 
dilatation  in  it,  not  participated  in  by  the  rest  of  the  glass.  If,  therefore,  the  stratum  of  molecules  A  B  were 
detached  from  the  rest  of  the  glass,  it  would  elongate  itself  so  as  to  project  at  its  two  ends  beyond  the  edges 
AC,  D  B.  When  the  heat  of  this  stratum  communicated  itself  to  the  next,  that  also  would  elongate  itself,  but 
in  a  less  degree ;  and  thus  after  a  very  long  time,  during  wJiich  the  heat  had  penetrated  to  the  farther  extremity 
of  the  glass,  its  outline  would  assume  the  form  a  C  D  6,  the  lines  a  C,  6  D  being  certain  curves  depending  on 
the  law  of  propagation  and  the  time  elapsed.  This  would  be  the  state  of  things  were  the  glass  plate  composed 


LIGHT.  565 

Light,      of  discrete  strata,  each  of  which  could  dilate  independently  of  all  the  rest.     And  since  in  each  of  these  (regarded     Part  IV. 
— v-*''  as  infinitely  thin)  the  temperature  and  strain  would  be  uniform,  there  would  arise  no  polarizing  action.     But,  in  s— v— ' 
reality,  the  case  is  quite  different ;  every  stratum  is  indissolubly  connected  along  its  whole  extent  with  the  strata 
adjacent,  and  can  neither  expand  nor  contract  without  forcing  them  to  participate  in  its  change  of  dimension. 
In  so  far,  then,  as  two  adjacent  strata  participate  in  the  change  of  temperature  they  expand  together;  but  when 
one  is  hotter  than  the  other,  the  former  is  found  to  expand  less,  and  the  other  more  than  if  they  were  inde- 
pendent.    Now  the  strain  thus  induced  on  any  stratum  is  not,  like  the  caloric  which  causes  it,  confined  by  the 
conducting  power  of  the  medium,  but  propagates  itself  instantly  (with  diminished  energy)  to  the  strata  beyond, 
by  reason  of  the  mutual  action  of  the  molecules. 

The  general  problem,  then,  to  investigate  the  actual  state  of  strain  of  any  molecule  at  any  moment  is  one  of      1098. 
some  complexity,  inasmuch  as  it  depends  at  once  on  the  laws  of  the  slow  propagation  of  heat,  and  the  instan-  State  of  th* 
taneous  but  variable  participation  of  change  of  figure  necessary  to  establish  among  the  particles  a  momentary  various  re 
equilibrium  under  the  circumstances  of  temperature  at  the  time  ;    but,  without  attempting  minutely  to  analyze  |>]°tnes  °s  toe 
the  effects,  if  we  content  ourselves  with  acquiring  a  general  idea  how  they  arise,  we  shall  find  little  difficulty.  strajn 
For  in  fig.  214,  if  we  conceive  the  stratum  A  B  b  a  adjacent  to  the  border  A  B  to  be  dilated  by  the  heat,  the  rest  determined, 
of  the  glass  retaining  its  original  temperature  ;  if  this  stratum  could  expand  separately,  its  edges  An,  B  6  would  f>g-  214. 
project  out  beyond  the  general  edges  C  a,  D  /3 ;    and  if  we  regard  two  terminal  strata  C  A  E  G,  D  B  F  II,  as 
detached  from  the  interior  portion  C  D  /3  a,  and  free  to  move  by  the  force  applied  at  their  extremities  A,  B,  they 
would  be  raised  by  the  dilatation  of  the  portion  A  B  6  a  into  the  situation  represented  in  the  figure,  turning 
round  C,  D  as  fulerums,  and  leaving  triangular  intervals  Caa,  D  ft  /3  vacant,  and  in  these  circumstances  there 
would  be  no  strain  on  any  part  of  the  system.     But  the  cohesion  of  the  glass  prevents  the  formation  of  these 
vacancies,  and  the  bars  or  levers  C  A  E  G,  D  B  F  H  cannot  move  into  this  situation  without  dragging  with  them, 
and  therefore  distending  the  strata  of  C  D  /3  «.     Let  P  Q  be  any  such  stratum,  and  let  it  be  distended  to  p  q. 
Then  by  its  elasticity  it  will  tend  to  draw  the  bars  C  A  E  G  and  B  D  H  F  together ;  and  its  action  will  therefore 
tend,  first,  to  produce  a  pressure  on  the  fulerums  C,  D,  urging  the  points  C  D  together,  and  therefore  bringing 
the  stratum  C  D  into  a  state  of  compression.     Secondly,  to  produce  also  a  pressure  on  A  a,  B  6,  or  a  resistance 
to  the  dilatation  of  A  B  ba,  which  its  increased  temperature  would  naturally  produce.     It  will  therefore  tend  to 
compress  back  the  strata  of  AB  ba  into  a  smaller  length  than  what  would  be  natural  to  them  in  their  heated 
state,  i.  e.  to  bring  them  also  into  a  relatively  compressed  state.     Thirdly,  the  tension  of  p  q  being  sustained  at 
C,  D  and  A,  B,  will  tend  to  bend  inwards  the  levers  A  C  G  E,  B  D  H  F,  rendering  them  concave  at  the  edges 
G  E,  H  F,  and  convex  at  C  A,   D  B,  and  thus  distending  the  lines  C  A,  D  B,  and  compressing  the  strata 
adjacent  to  E  G,  H  F. 

From  this  reasoning  it  is  clear,  that  the  glass,  in  consequence  of  these  various  strains,  will  assume  a  figure      1099. 
concave  on  all  its  edges,  but  chiefly  so  at  the  lateral  ones  A  C,  D  B,  as  in  fig  215  ;  and  that  the  state  of  strain  Production 
of  its  various  parts  will  be  as  there  expressed,  all  the  edges  being  compressed,  but  principally  AB  and  C  D,  and  of  fringes  of 
the  interior  distended.     The  limit  between  the  distended  and  compressed  portions  parallel  to  A  B  must  neces-  "{[j^ters 
sarily  be  marked  by  neutral  lines  a  b,  c  d  on  either  side  of  which  the  strain  will  increase,  being  a  maximum  in  fjg-  215. ' 
the  middle  and  on  or  near  the  edges.     Consequently,  it  ought  to  polarize  four  sets  of  fringes,  having  a  b,  c  d 
for  their  origins,  and  of  which  the  two  external  (or  those  between  these  lines  to  the  edge)  ought  to  have  a 
character  opposite  to  those  of  the  internal,  the  portion  of  the  intromitted  pencil  polarized  parallel  to  A  B  being 
propagated  faster  than  that  parallel  to  A  C  in   the  one   case,   and  slower  in  the  other.     This   opposition  of 
characteis  is  conformable  to  Dr.  Brewster's  observations,  who   states  (PA/7.  Trans.  1816)  that  the  parts  of  the 
glass  which  exhibit  the  two  exterior  sets  of  fringes  (adjacent  to  the  edges  A  B,  C  D)  have  "  the  structure  of" 
attractive  crystals,  while  the  parts  which  exhibit  the  interior  and  terminal  sets  have  that  of  repulsive  ones ; 
meaning,  of  course,  in  the  Lmguage  of  the  undulatory  doctrine,  that  the  order  of  velocities  of  the  doubly 
refracted  pencils  is  reversed  in  passing  from  one  region  of  the  glass  to  the  other,  for  of  its  actual  structure  we  • 
can  know  nothing.     That  the  terminal  fringes  ought  (as  observed)  to  have  the  same  character  as  the  interior  is  The  termi- 
a  necessary  consequence  of  the  above  reasoning,  for  the  terminal  regions  D  B,  AC  are  compressed  in  directions  na'  ff'n6e* 
para/Id  to  their  edges,  and  therefore  perpendicular  to  the  direction  in  which  the  central  portion  is  distended ; 
and  we  have  already  seen  that  compression  in  one  direction  is  equivalent  (so  far  as  the  character  of  the  tints 
produced  is  concerned)  to  distension  in  that  perpendicular  to  it. 

Lastly,  the  black  lines  separating  the  terminal  fringes  from  the  interior  ones,  arise  from  the  combined  action      1100. 
of  the  tension  of  the  interior  region  parallel  to  A  B  (fig.  214)  exerting  itself  on  any  point  as  q  on  the  inner  Neutral 
border  of  the  terminal  portion  D  B  F  H,  (which  we  have  regarded  as  an  elastic  bar,  or  lever,)  and  the  distension  l>nes  sepa- 
of  the  line  D  B  also  exerting  itself  at  q,  and  arising  from  the  convexity  given  to  this  line.     In  virtue  of  these  raunS  aclJa 
two  forces,  every  point  q  in  a  certain  line  at  a  proper  distance  from  the  extreme  edge  H  F,  will  be  equally  y"™' S 
distended  in  opposite  directions,  and  will  therefore  be  in  a  neutral  state,  as  to  polarization,  and,  of  course, 
appear  black.     The  terminal  fringes  are  less  developed  than  the  rest,  because  they  arise  simply  from  the  flexure 
of  the  edges  H  F,  G  E,  which  is  an  indirect  elFect  of  the  principal  force,  and  is  very  small,  (owing  to  the  small 
dilatability  of  glass  by  heat,  and  consequent   minuteness  of  the  versed  sine  of  the  curve  into  which   they  are 
distorted,)  and  the  line  of  indifference  separating  them  from  the  others  lies  near  the  edges ;  for  the  same  reason, 
the  tension  of  the  convex   line  D  B  being  small,  and  therefore  putting  itself  in  equilibrium  with  that  of  the 
distended  column  p  q  at  a  point  q  near  its  extremity,  where  it  is  evident  that  the  strain  parallel  to  p  q  must  be 
much  diminished  ;  the  greater  portion  of  the  whole  tension  of  p  q  being  resisted  by  the  spring  of  laminae  situated 
slill  further  from  the  edge  than  D  B. 

If  a  lamina  of  glass,  uniformly  heated,  be  suddenly  cooled  at  one  of  its  edges,  the  reverse  of  all  these  effects      1101 
will  arise;    the  outer  column  ABaft  (fig.  214)  will  suddenly  contract  and  compiess  violently  the  columns 


566  LIGHT. 

l.Ulit.      beyond  a  /3,   from  which  no  heat  has  yet  been  abstracted,  and  drag  inwards  the  ends  of  the  terminal  levers    P,irt  IV. 
'  — ^s~— J  E  A  G  C,  B  F  II  D,  which  will  thus  be  violently  pressed  on  the  parts  ft  Q  and  a  P  as  fulcra ;  and  their  action  ^— v— — 
Phenomena  being  thus  transmitted  to  the  opposite  edge  C  D  will  tend  to  lengthen  it,  and  thus  bring  it,  as  well  as  the  edge 
^ass'rec'tan-  ^  ®'  mto  &  Distended  state.     The  terminal  edges  will  also  be  sprung  outwards.     The  strain  on  every  point 
"le  cooled     w'"  ^e  exactly  the  reverse  of  what  is  expressed  in  fig.  215,  and  a  corresponding  inversion  of  the  characters 
at  one  edge,  of  the  tints  will  take  place  ;  all  which  is  agreeable  to  Dr.  Brewster's  observation,  (Prop.  14  of  the  Memoir  cited.) 

1102.  When  a  crack  takes  place  in  a  piece  of  unequally  heated  glass,  the  directions  and  intensities  of  the  straining 
Effect  of  a    forces  in  every  part,  which  depend  wholly  on  the  cohesion  of  its  molecules,  and  the  continuity  of  the  levers, 

springs,  &c.  into  which  it  may  be  mentally  conceived  to  be  divided,  is   suddenly  altered  ;    and  the  fringes  are 
accordingly  observed  to  take  instantly  a  new  arrangement,  and  assume  forms  related  to  the  ligure  of  that  part 
of  the  glass  which  preserves  its  continuity.     To  analyze   the  modifications   arising  from  variations  of  external 
figure  and  different  applications  of  the  heat,  would  be  to  involve  ourselves  unnecessarily  in  a  wilderness  of  com- 
plexity.    One  simple  case  may,  however,  be  noticed,  in  which  the  centre  of  a  circular  piece  of  glass  is  heated. 
Each  exterior  an  mil  us  of  this  will  be  placed  in  a  state  of  distension  parallel  to  its  circumference,  and  will 
circular  *      compress  all  within  it  by  a  force  parallel  to  the  radius.     The  central  point  will  be  neutral,  being  equally  confined 
plate  heated  'n   a"   directions,  and  the   annuli  adjacent  to  the  centre  will  in  like  manner  be  compressed  both  radially  and 
in  the  circumferentially.     The  radial  strain  continues  as  we  recede  from  the  centre,  but  the  circumferential  diminishes, 

centre.  and  at  length,  as  already  said,  changes  to  a  state  of  distension,  and  of  course  passes  through  a  neutral  state, 
thus  giving  rise  to  a  black  circle  and  concentric  fringes  of  opposite  characters,  the  whole  of  which  will  be  inter- 
sected by  the  arms  of  a  black  cross  parallel  and  perpendicular  to  the  plane  of  primitive  polarization,  and  which 
of  course  remains  fixed  while  the  plate  is  turned  round  in  its  own  plane. 

1103.  There  is  only  one  experiment  of  Dr.  Brewster  which  seems  hostile  to  the  theory  here  stated.     He  made  a 
Singular       partial  crack  with  a  red-hot  iron  in  a  very  thick  piece  of  glass,  and  allowed  it  to  close  by  long  standing,  which 

fleet  of  a  jt  jjj^  go  as  j.Q  disappear  entirely.  In  this  state,  the  glass,  when  unequally  heated,  exhibited  the  same  fringes, 
allowed  to  as  if  no  crack  had  existed  ;  but  the  moment  the  crack  was  opened  by  a  slight  heat  applied  near  it,  they  suddenly 
close.  changed  their  figure,  and  assumed  that  due  to  the  portion  having  the  crack  for  a  part  of  its  outline.  It  seems, 

however,  that  a  very  great  adhesive  force  takes  place  between  the  surfaces  of  glass  when  thus  in  optical  contact ; 
and  to  those  who  are  aware  how  the  free  expansion  and  contraction  of  dissimilar  metallic  bars  may  be  com- 
manded, and  the  bars  in  consequence  made  to  ply  on  change  of  temperature  by  mere  forcible  juxtaposition, 
without  soldering,  till  the  difference  of  expansion  has  reached  a  certain  point,  when  they  give  way  with  a  snap 
and  regain  their  state  of  equilibrium,  the  anomaly  will  not  appear  in  the  light  of  a  radical  objection.  (We 
think  it  not  improbable,  that  the  musical  sounds  said  to  issue  at  sunrise  from  certain  statues,  may  originate  in  some 
pyrometrical  action  of  the  kind  here  alluded  to.  We  have  often  been  amused  by  a  similar  effect  produced  in  the 
bars  of  the  grate  of  a  jire-place.) 

1104.  Such  are,  in  general,  the  transient  effects  of  a  heat  below  the  softening  point  of  glass,  unequally  distributed 
Phenomena  through  its  substance.     But  if  a  mass  of  glass  be  heated  up  to,  or  beyond  that  point,  so  as  to  allow  its  mole- 
of  unan-       cules  to  glide  with  more  or  less  freedom  on  one  another,  and  adapt  themselves  to  any  form  impressed  on  the 
nealed  glass  masS)  ancj  ^en  suddenly  cooled,  either  by  plunging  into  water,  or  by  exposure  to  cold  air,  the  heat  is  abstracted 

from  its  external  strata  with  so  much  greater  rapidity  than  it  can  be  supplied  by  conduction  from  within,  that 
they  become  rigid,  while  the  inner  portions  are  still  soft  and  yielding.  At  this  instant,  there  is  therefore  no  strain 
in  any  part  ;  but,  the  abstraction  of  the  heat  still  going  on,  the  internal  parts  at  length  become  solid,  and  tend, 
of  course,  to  contract  in  their  dimensions.  In  this,  however,  they  are  prevented  by  the  external  crust  already 
formed,  which  acts  as  an  arch  or  vault,  and  keeps  them  distended,  at  the  same  time  that  these  latter  portions 
themselves  are  to  a  certain  extent  forced  to  obey  the  inward  tension,  and  are  strained  inwards  from  their  figure 
of  equilibrium.  Glass  in  this  state  is  said  to  be  unannealed.  If  the  cooling  has  been  sudden,  and  the  mass 
considerable,  it  either  splits  in  the  act  of  cooling,  or  flies  to  pieces,  when  cold,  spontaneously,  or  on  the  slightest 
scratch  which  destroys  the  continuity  of  its  surface ;  and  the  pieces  when  put  together  again  (which,  however, 
is  seldom  practicable,  as  it  usually  flies  into  innumerable  fragments,  or  even  to  powder,  as  is  familiarly  shown 
Rupert's  in  the  glass  tears  called  Rupert's  drops,  which  exhibit  a  very  high  polarizing  energy  from  their  intense  strain, 
drops.  an(i  which  burst  with  a  violence  amounting  to  explosion,  on  the  rupture  of  their  long  slender  tails)  are  found  not 

to  fit,  but  to  leave  a  slight  vacancy ;  thus  satisfactorily  proving  the  state  of  unnatural  and  violent  distension  in 
which  its  interval  parts  have  been  held.  The  case  is  precisely  analogous  to  that  of  a  gelatinous  substance 
allowed  to  indurate  under  the  influence  of  dilating  forces.  (See  Art.  1094.) 

1105.  If  the  cooling  be  less  sudden,  and  carefully  managed,  the  glass,  though  much  more  brittle   than  ordinary 
Patterns       annealed  glass,  is  yet  susceptible  (with  great  caution)  of  being  cut  and  polished  ;  and  in  this  state,  if  polarized 
exnibited      \\g\\l  be  passed  through  it,  it  exhibits  coloured  phenomena  of  astonishing  variety  and  splendour,  forming  fringes, 
s  ua'reana   irises,  and  patterns  of  exquisite  regularity  and  richness,  according  to  the  form  and  size  of  the  mass,  and  the 
rectangular  degree  of  strain  to  which  it  is  subjected.     In  all  these  cases  if  the  external  form  be  varied,  the  pattern  varies  cor- 
unannealed    respondingly,  as  it  is  easy  to  perceive  it  ought ;  for  if  any  part  of  tile  exterior  crust  be  removed,  that  part  of  the 
plates,          strain  which  it  sustained  will  fall  on  the  remainder,  and  on  the  new  surface  produced.     Figures  216,  217,  and 
Fl|2^16  ~   218,  represent  the  patterns  exhibited  by  a  circular,  a  square,  and  a  rectangular  plate  of  about  J-  inch  thick, 

the  two  latter  being  placed  so  as  to  have  one  side  parallel  to  the  plane  of  primitive  polarization.  Figure  219 
and  220  represent  the  patterns  shown  by  the  two  latter  in  azimuth  45°,  and  fig.  221  that  arising  from  the 
crossing  of  two  plates  equal  and  similar  to  fig.  220,  each  being  in  azimuth  45°.  In  all  these  cases  the  laws  of 
superposition  of  Art.  1089  are  observed,  when  similar  points  of  similar  plates  are  laid  together.  If  symme- 
trically, the  tints  polarized  is  the  same  as  would  be  polarized  by  one  plate  whose  thickness  is  their  sum  ;  if 
crosswise,  their  difference. 


LIGHT.  5U7 

Light.          If  a  square  or  rectangular  plate  be  turned  about  in  its  own  plane,  from  azimuth  O3,  the  arms  of  the  black    Part  IV 
•— -y— •»'  cross  dividing  it  into  four  quarters  become  curved,  as  in  fig.  222,  and  pass  in  succession  over  every  part  of  the  v— — v- — '' 
disc  ;  thus  showing  that  the  positions  of  the  axes  of  elasticity  of  the  molecules  vary  for  every  different  point  of      1106. 
the  plate,  and  in  different  parts  of  it  have  every  possible  situation.     We  shall  not  here  attempt  to  analyze  the  Effect  ot 
mechanical  state  of  the  molecules  in  any  case,  as  it  would  lead  us  too  far  ;    but  merely  mention  an  experiment  '"™j?an 
of  Dr.  Brewster,  which  is  sufficient  to  show  the  conformity  of  our  theory  of  these  figures  with  fact.     According  unannealed 
to  this  excellent  observer,  the  fringes  parallel  to  the  edge  A  B  of  the  rectangle  (fig.  220)  are  similar  in  their  plate  in  its 
character  to  those  produced  by  setting  the  corresponding  edge  of  a  similar  plate  of  annealed  glass  on  a  hot  iron.  ow"  P1™6- 
Now,  in  the  latter  case,  the  exterior  fringes  adjacent  to  A  B,  C  D  arise  from  a  compressed  state  of  the  columns  Flg-  2- 
parallel  to  AB;  and  the  interior,  from  a  distended.     And,  in  the  unannealed  plate  the  distribution  of  the  forces  Relation  of 
is  almost  exactly  similar  to  that  described  in  Art.  1098  and  1099.     In  fact,  such  a  plate  maybe  likened,  in  some  these  phe- 
respects,  to  a  frame  of  wood  over  which  an  elastic  surface  is  stretched  like  a  drum.     The  four  sides  will  all  be  nomena  to 
curved  inwards  by  its  tension,  and  they  will  all  be  compressed  in  the  direction  of  their  length  by  the  direct  tl!^gjgnt]i 
tension,  independent  of  the  secondary  effect  produced  by  their  curvature.     The  terminal  fringes  in  the  articles  heated 
referred  to  arise  solely  from  the  secondary  forces  thus  developed  ;   but  the  analogy  between  the  cases  would  be  annealed 
complete,  if,  instead  of  supposing  the  annealed  plate  heated  at  one  edge  only,  the  heat  were  applied  at  all  the  plates. 
four  simultaneously,  by  surrounding  it  with  a  frame  of  hot  iron.     For  a  farther  account  of  the  beautiful  and 
interesting  phenomena  produced  by  unannealed  glass,  we  must  refer  the  reader  to  Dr.  Brewster's  curious  Paper 
already  cited. 

M.  Fresnel  has  succeeded  in  rendering  sensible  the  bifurcation  of  the  pencils  produced  by  glass  subjected  to      1107. 
pressure,  by  an  ingenious  combination  of  prisms  having  their  refracting  angles  turned  opposite  ways,  and  of  which 
the  alternate  ones  are  compressed  in  planes  at  right  angles  to  each  other,  thus  (as  in  the  case  of  the  double 
refraction  along  the  axis  of  quartz)  doubling  the  effect  produced. 

The  effects  produced  by  unequal  heat  and  pressure  on   crystallized  bodies,  in   altering  their  relations  to  light      1108. 
transmitted  through  them,  are  less  sensibly  marked  than  in  uncrystallized,  being  masked  by  the  more  powerful  Effects  of 
effects  produced  by  the  usual  doubly  refractive  powers.     In  crystals,  however,  where  these  powers  are  feeble,  or  unequal 
in  which  they  do  not  exist  in   any  sensible  degree,  fas  in   fluor  spar,  muriate  of  soda,  and  other  crystals  which  he 
belong  to  the  tessular  system,  Dr.  Brewster  has  shown  that  a  polarizing  and  doubly-refractive  action  is  deve-  crystallized 
loped  by  these  causes  just  as^n  uncrystallized  ones  ;  and  M.  Biot,  by  applying  violent  pressure  to  crystallized  bodies. 
substances  while  viewing  through  them  their  systems  of  rings  in  the  immediate  vicinity  of  their  axes  where  the 
polarizing  action  is  very  weak,  has  succeeded  in  producing  an  evident  distortion  of  the  rings  from  the  regularity 
of  their  form,  thus  rendering  it  manifest,  that  it  is  only  the  extreme  feebleness  of  the   polarizing  action   so 
induced  in  comparison  with  the  ordinary  action   of  the  crystal,  which  prevents  its  becoming  sensible   in   all 
directions. 

In  applying  what  is  here  said  to  heat,  however,  we  consider  only  its  indirect  action,  or  that  arising  from  its      1109.    • 
unequal  distribution,  inducing  a  strain,  and  thus  resolving  itself  into  pressure,  as  above  shown.     But  Professor  Mitscher- 
Mitscherlich  in  a  most  interesting  series  of  researches  (which  we  hope,  ere  long  to  see  embodied  in  a  regular  l|ch's  re- 
form, but  of  which  at  present  only  the  most  meagre  and  imperfect  details  have  reached  us)  has  shown  that  the  th'^dirata"1 
action  of  heat  on  crystallized  bodies,  even  when  uniformly  distributed,   so  that  the  whole  mass  shall  be  at  one  tion  of 
and  the  same  temperature,  is  totally  different  from  what  obtains  in  uncrystallized,     In  the  latter  (as  well  as  in  crystals  by 
crystals  of  the  tessular  system)  an  elevation  of  temperature,  common  to  the  whole  mass,  produces  an  equal  dila-  t>eat- 
tation  in  all  directions,  the  mass  merely  increases  in  dimensions,  without  change  of  figure.     In  crystals,  however, 
not  belonging  to  the  tessular  system,  i.  e.  whose  forms  are  not  symmetrical  relative  to  three  rectangular  axes, 
the  dilatation  caused  by  increase  of  temperature  is  so  far  from  being   the  same  in  all  directions,  that  in  some 
cases  a  dilatation  in  one  direction  is  accompanied  with  an  actual  contraction  in  another. 

Of  this  important  fact,  (the  most  important,  doubtless,  that  has  yet  appeared  in  pyrometry,)  M.  Mitscherlich      1110. 
has  adduced  a  remarkable  and  striking  instance  in  the  ordinary  Iceland  spar,  (carbonate  of  lime.)     This  sub-  Pyrometri- 
stance  when  heated,  dilates  in  the  direction  of  the  axis  of  the  obtuse  rhomboid  which  is  the  primitive  form  of  its  ^Isp0rf011'ceer 
crystals,  and  contracts  in  every  direction  at  right  angles  to  that  axis,  so  that  there  must  exist  an  intermediate  [^"spaT" 
direction,  in  which  this  substance  is  neither  lengthened  nor  contracted  by  change  of  temperature.     A  necessary 
consequence  of  such  inequality  of  pyrometric  action  is,  that  the  angles  of  the   primitive  form  will  undergo  a 
variation,  the  rhomboid  becoming  less  obtuse  as  the  temperature  increases,  and  this  has  been  ascertained  to  be 
the  case  by  direct  measurement ;   M.  Mitscherlich  having  found,  that  an   elevation  of  temperature  from  the 
freezing  to  the  boiling  point  of  water  psoduced  a  diminution  of  8' 30"  in  the  dihedral  angle  at  the  extremities  of 
the  axis  of  the  rhomboid,  (Bulletin  des  Sciences  publie  par  la  Societe  Philomatique  de  Paris,  1824,  p.  40.) 

M.  Mitscherlich  assured  himself  of  the  fact  in  question  by  direct  measurement  of  a  plate  of  Iceland  spar     1111. 
parallel  to  the  axis,  at  different  temperatures,  by  the  aid  of  the  "  Spherometer,"  a  delicate  species  of  calibre  con-  Mo<Ie  of . 
trived  by  M.  Biot  for  measuring  the  thickness  of  any  laminar  solid  by  the  revolution  of  a  screw  whose  point  is  (^taln'"s 
just  brought  into  light  contact  with  the  surface,  and  by  which  the  10,000th  of  an  inch  is  readily  appreciated  and 
measured.     The  experiment  is  necessarily  one  of  great  delicacy,  but  our  readers  may  assure  themselves  at  least 
of  the  general  fact  of  unequal  change  of  dimension  by  change  of  temperature,  by  a  very  simple   experiment 
requiring  almost  no  apparatus.     Let  a  small  quantity  of  the  sulphate  of  potash  and  copper,  (an  anhydrous  salt  Pyrometri- 
easily  formed  by  crystallizing  together  the   sulphates  of  potash  and  of  copper,)  be  melted  in  a  spoon  over  a  cal  property 
spirit  lamp.     The  fusion  takes  place  at  a  heat  just  below  redness,  and  produces  a  liquid  of  a  dark  green  colour.  °ff  s^t^te 
The  heat  being  withdrawn,  it  fixes  into  a  solid  of  a  brilliant  emerald  green   colour,  and  remains  solid  and  Jnl^opper 
coherent  till  the  temperature  sinks  nearly  to  that  of  boiling  water,  when  all  at  once  its  cohesion  is  destroyed  ;  a 
commotion  takes  place  throughout  the  whole  mass,  beginning  from  the  surface,  each  molecule,  as  if  animated. 


568  LIGHT. 

Light.      starting  up  and  separating  itself  from  the   rest,  till,  in  a  few  moments,  the   whole  is  resolved  into  a  heap  of     Part  IV. 
v— - s~~-  incoherent  powder,  a  result  which  could  evidently  not  take  place,  had  all  the  minute  and  interlaced  crystals  of  v— •^-~"- 
which  the  congealed  salt  consisted  contracted  equally  in  all  directions  by  the  cooling  process,  as  in  that  case 
their  juxtaposition  would  not  be  disturbed.     Phenomena  somewhat  similar,  and  referable  to  the  same  principles, 
have  (if  we  remember  right)  been  encountered  by  M.  Achard  in  the  fusion  of  various  frits  for  glasses,  &c. 

1112  The  relation  of  the  optical  and  crystallographical  characters  of  bodies  is  so  intimate,  that  no  change  can  be 
Double  re-  supposed  to  take  place  in  the  latter  without  a  corresponding  alteration  in  the  former.     As  the  rhomboid  of  Ice- 
fraction  of    land  spar  becomes  less  obtuse   by  heat,  and  therefore  approximates  nearer  to  the  cube,  in  which  the  double 
crystals  ya-  refraction  is  nothing,  it  might  be  expected  that  the  power  of  double  refraction  should  diminish,  and  this  result 

ile  WHO   nas  jjeell   verifiefi   by  M.  Mitscherlich  by  direct  measurement.     More  recently,  the  same  distinguished  chemist 

and  philosopher  has  ascertained  the  still  more  remarkable  and  striking  fact,  that  the  ordinary  sulphate  of  lime 

property  of6  or  ffyPsum>  which,  at  common  temperatures,  has  two  optic  axes  in  the  plane  of  its  lamina:,  inclined  at  60°  to 

sulphate  of  each  other,  undergoes  a  much  greater  change  by  elevation  of  temperature  ;  the  axes  gradually  approaching  each 

lime.  other,  collapsing  into  one,  and  (when  yet  further  heated)  actually  opening  out  again  in  a  plane  at  ris*lit  angles 

to  the   lamiiue,    thus  affording    a  beautiful    exemplification  of  Fresnel's  theory  of  the   optic  axes   as  above 

explained. 

1113  This  singular  result  we  cite  from  memory,  having  in  vain  searched  for  the  original  source  of  our  information  ; 
but  it  might  have  been  expected,  from  the  low  temperature  at  which  the  chemical  constitution  of  this  crystal  is 
subverted,  by  the  disengagement  of  its  water,  that  the  changes  in  its  optical   relations  by  heat  would  be  much 
more  striking  than  in  more  indestructible  bodies.     We  have  not,  at  this  moment,  an  opportunity  of  fully  verify- 
ing the  fact ;   but  we  observe,  that  the  tints  developed  by  a  plate  of  sulphate  of  lime  now  before  us,  exposed  as 
usual  to  polarized  light,  rise  rapidly  in  the  scale  when  the  plate  is  moderately  warmed  by  the  heat  of  a  candle 
held  at  some  distance  below  it,  and  sink  again  when  the  heat  is  withdrawn,  which,  so  far  as  it  goes,  is  in  con- 
formity with  the  result  above  stated.     Mica,  on  the  contrary,  similarly  treated,  undergoes  no   apparent  change 
in   the  position  of  its  axes  or  the  size  of  its  rings,  though  heated  nearly  to   ignition.     The  subject  is  in  the 
highest  degree  interesting  and  important,  and  lays  open  a  new  and  most  extensive  field  for  optical  investiga- 
tion.    It  is  in  excellent  hands,  and  we  doubt  not  will,  ere   long,  form  a  conspicuous  feature  in  the  splendid 
series  of  crystallographical  discovery  which  has  already  so  preeminently  distinguished  its  author 

§  XIII.     Of  the  Use  of  Properties  of  Light  in  affording  Characters  for  determining  and  identify  ins;  Chemical 
and  Mineral  Species,  and  for  investigating  the  intimate.  Constitution  and  Structure  of  Natural  Bodies. 

'  1114.         Newton,  who  "  looked  all  nature  through,"  was  the  first  to  observe  a  connection  between  the  refractive  powers 
Relation       of  transparent  media  and  their  chemical  properties.     His  well  known  conjecture  of  the  inflammable  nature  of 
between  the  the  diamond,  from  its  high  refractive  power,  so  remarkably  verified  by  the  subsequent  discovery  of  its  one  and 
^wers'Tnd  on^  chemical  constituent,  (carbon,)  was,  perhaps,  less  remarkable  for  its  boldness,  at  a  period  when  Chemistry 
chemical"     consisted  in  a  mere  jargon,  in  which  salt,  sulphur,  earth,  oil,  and  mercury  might  be  almost  indifferently  substi- 
composition  tuted  for  one  another,  than  it  would  have  been  fifty  years  later.     His  divination  of  the  inflammable  nature   of 
of  bodies,     one  of  the  constituents  of  water  is   at  least  equally  striking  as  an  instance  of  sagacity,  and  even  more  remark- 
able, for  the  important  influence  which  its  verification  has  exercised  over  the  whole  science  of  Chemistry.    These 
instances  suffice  to  show  the  value  of  the  refractive  index,  either  taken  in  conjunction  with  the  specific  gravity 
of  a  medium,  or  separately  as  a  physical   character.      The  refractive  indices  of  a  vast  variety  of  bodies  have 
been  ascertained  by  the  labours  of  Newton  and  later  experimenters,  among  whom  Dr.  Brewster  and  Dr.  Wol- 
laston  have  been  the  largest  contributors  to  our  knowledge.    They  may  be  grouped  together  in  a  general  way, 
in  order  of  magnitude,  as  follows : 

ijit          Class  1.  Gases  and  vapours.     Refractive  index  from  1.000  to  1.002,  under  ordinary  circumstances  of  pressure 
Classified-    and  temperature. 

tionofbo-        Class  2.     /x  =r  1.05  ....    fi  —  1.45.     Comprising  the  condensed  gases;    ethereal,  spirituous,  and  aqueous 
dies  accord-  liquids  ;  acid,  alkaline,  and  saline  solutions,  (not  metallic.) 

ing  to  their       Class  3.    Comprising,  first,  almost  all  unctuous,  fatty,  waxy,  gummy,  and  resinous  bodies  ;  camphors,  balsams, 
d* n-'ties       vegetable  and  animal  inflammables,  and  all  the  varieties  of  hydro-carbon.     Secondly,  stones  and  vitreous  com- 
pounds, in  which  the  alkalis  and  lighter  alkaline  earths  in  combination  with  silica,  alumina,  &c.  are  the  predo- 
minant ingredients.     Thirdly,  saline  bodies  not  having  the  heavy  metals,  or  the   metallic  acids  predominant 

ingredients,  p,  =  1.40 l.fiO. 

Class  4.  Pastes,  (glasses  with  much  lead,)  and,  in  general,  compounds  in  which  lead,  silver,  mercury,  and  the 
heavy  metals,  or  their  oxides  abound.  Precious  stones,  simple  combustibles  in  the  solid  state,  including  the 
metals  themselves. 

p  =  1.60  and  upwards. 

These  classes,  however,  admit  of  so  many  exceptions  and  anomalies,  and  are  themselves  so  vague  and  indefinite, 
that  we  shall  not  attempt  to  distribute  the  observed  indices  under  any  of  them,  but  rather  prefer,  for  conve- 
nience of  reference,  presenting  the  whole  list  in  the  form  of  a  Table,  arranged  in  order  of  magnitude,  in  which 
all  these  classes  are  mingled  indiscriminately — a  form,  in  some  measure,  consecrated  by  usage. 


L  I  G  H  T. 


569 


Light 


Table  of  Refractive  Indices,  or  Values  of  p.  for  Rays  of  Mean  Refrangibility,  (unless  expressed  to  the  contrary.) 
Dr.  Wollaston's  results,  however,  are  all  (according  to  Dr.  Young,  Philosophical  Transactions,  vol.  xcii. 
p.  370,)  to  be  regarded  as  belonging  to  the  Extreme  Red  Rays. 


Part  IV. 


1116. 


N.  B.   In  this  Table  the  authorities  are  referred  to  as  follows : 


Br    Brewster,  Encyclop.  Ed,  and  Treatise  on  New  Philosophical  Instrument!.  Bos.  Boscovich. 

B  Y.  Dr.  Young's  Calculations  of  Dr.  Brewster's  Unreduced  Observations.    Quarterly  Journal,  vol.  xxii. 
Bi.  Biot.  F.  Faraday.  Du.  Dulonsr.  M.  Mains.  N.  Newton.  Fr.  Fraunhofer. 

w!  \Vollaston,  Phil.  Trans.  He.  From  our  own  observation.  Eul.  Kuler  the  younger. 

C.  and  H.,  authorities  cited  by  Dr.  Young  in  his  Lectures. 


Vacuum  1.000000 

Vitreous  humour  of  the  haddock  

....      1.3394     Br. 

GASES, 
at  the  freezing  temperature  and  pressure  =  291".922  =  0-.76 
Hydrogen  000138     Du. 

Ditto      

1  .340       B.Y. 

1  353       B  Y 

1  339       B  Y 

1.339       B.Y. 

Oxygen     000272     Du. 
Atmnsnheric  air                                              000294     Bi. 

Azote     009300     Du. 
Nitrous  gas  000303     Du. 
Carbonic  oxide                          000340     Du. 

Vinegar  (distilled)  
Ditto      

'"  \l  .349  J  Br- 
...      1.344       Eul. 
....      1.372       H. 

Ammonia  000385     Du. 
Carburetted  hydrogen  000443     Du. 
Carbonicacid  000449     Du. 
Muriatic  acid    000449     Du. 

Vinegar     
Acetic  acid  (  ?  strength)  
Jelly  fish  (Medusa  jEquora)     
White  of  egg    

1.347       B.Y 
1.S96       Br. 
....     1.345       Br. 
1.351       Eul. 
1.351       B.Y. 

Hydrocyanic  acid  1.000451     Du. 
Nitrous  oxide  1.000503    Du. 
Sulphuretted  hydrogen    1.000644     Du. 

Human  blood    
Saturated  aqueous  solution  of  alum    

.  ...      1.354       B.Y. 
1.356       He. 
1356       B.Y 

Sulphurous  acid    1.000665     Du. 
defiant  gas                                          1.000678     Du. 

Ether   .            

f  1.358       W. 

Chlorine    1.000772     Du. 

•"  1  1.374       B.Y. 
1  360       W 

Protophosphuretted  hydrogen  1.000789     Du. 
Cyanogen                               1.000834     Du. 

J1361       Br. 

Muriatic  ether  1.001095     Du. 
Phoseen                                                                  1001159     Du. 

Brandy  

"••   71.359      B.Y. 
1.360       B.Y. 

Vapour  of   sulphuric   ether  (boiling  point   at 
35°centig.)  1.001530     Du. 

....     1.360       B.Y. 
f  1.368       Br. 

"•   \1.379       B.Y. 

Ditto  (S   G   0  866) 

1  370       N 

LIQUIDS  AND  SOLIDS. 
ftSS 

Ether  expanded  by  heat  to  three  times  its  volume     1.0570     Br. 
Tabasheer  from  Vellore,  1  yellowish  transparent 

Ditto                      ...                               .. 

1  371       C 

Ditto  (rectified  spirits)     

1.372       He. 

1  374       Br 

.  .  ..     1  377       B  Y. 

1  375       C 

First   new  fluid   discovered  by   Dr.  Brewster  in 

Ditto  (SGI  134)  

1  392       He 

1  395       B  Y 

Tabasheer,  transparent,  from  Nagpore    1.1454     Br. 

1  401       Br 

Ditto                 ditto         ditto       another  specimen     1.1503     Br. 
Ditto,  whitest  variety,  from  Nagpore  1.1825     Br. 

.  .  .  .     1  4098     Bi 

1  379       B  Y 

New    Hnid  discovered  by  Dr.  Brewster  in  ame- 
thyst, at  83J°  Fahr  1.2106     Br. 

.  .  .  .      1  384       He 

Second  new  fluid  discovered  by  Dr.  Brewster  in 
topaz,  at  83°  Fahr  1.2946     Br. 

1  395       Br 

Pus 

1  395       B  Y 

..,.„..                                                      f     much  less     \  P 

Nitrous  oxide  liquefied  by  pressure     \  than  water   jr. 

f  1.396       Br. 

Muriatic  acid  gas    ditto  ditto  j  ..nearl            ,     J  «S=h    }p. 
Carbonic  acid  gas  ditto  ditto  J                               I     »«*"       > 
(    1.307      Br. 
Ice          ..               {     1.3085     Br. 

'••   11.404       B.Y. 
1  406       Br 

Crystalline  lens  of  the  eye  (human?)  outer 
Ditto                               ditto                  middle 
Ditto                               ditto                  centre 

coat     1.3767     Br. 
coat     1.3786     Br. 
....      1.3990     Br. 
1  386      B  Y 

(     1.3100     W. 

f     rather  lesi      1  p. 

Ditto              ditto            middle  coat  

...      1.428       B.Y. 

....     1.436       B.Y. 

Ditto                 ditto                                               ..      1  316       Br 

Ditto              ditto                 middle  coat  .... 

OIL                        -  ,  ..          f.    j   i                                                          f        equal  to      \-e 

Ditto              ditto                 centre   

....      1.439       B.Y. 

Sulphurous  acid  liquefied  by  pressure  \      water      /*• 
(N. 
Water      U36      MV. 

Ditto  of  the  ox     

1  1.3801 

Ditto      ditto     ...        . 

'   11.447J        • 

(Br. 
Sulphuretted  hydrogen  liquefied  by  pressure   .  .  {fj££r£5r*r}  F- 

(greater  than  j 
^c»ttr  "han  }F. 
all  the  other 
liquefied  gatesj 

Ditto  of  the  pigeon  

1  406       B  Y 

..         1  403       B  Y 

Solution  of  potash,  S.  G.  1.416,  (ray  El     .. 

...      1  .40563   Pr 

Nitric  acid  (S.  G.  1.48)  

/  1.410       B.Y. 

'    {1.410      W. 
1  412      C 

Ditto  of  the  haddock    1.341       BY. 

Vitreous        ditto  1.336       W. 

1  4^6          Rr 

VOL.  IV.                                                                                                                                                    4.J5 

570 


L  I  G  H  T. 


Light. 


.     1.426 

B.Y. 
B.Y. 
N. 
He. 
W. 
Br. 
W. 
Br. 
Br. 
B.Y. 
B.Y. 
B.Y. 
Br. 
W. 
B.Y. 
C. 
Br. 
B.Y. 
Br. 
B.Y. 
Br. 
W. 
B.Y. 
W. 
N. 
B.Y. 
W. 
B.Y. 
Br. 
B.Y. 
Br. 
N. 
C. 
Br. 
W. 
B.Y. 
Br. 
B.Y. 
Br. 
B.Y. 

W. 

B.Y. 

Br. 
W. 
N. 
Br. 
B.Y. 
W. 
C. 
B.Y. 
He. 
Fr. 
N. 
W. 
Br. 
He. 
B.Y. 
Br. 
B.Y. 
Br. 
B.Y. 
Br. 
Br. 
B.Y. 
W. 
B.Y. 

B.Y. 

Br. 

B..Y. 

W. 
N. 
Br. 
Br. 
B.Y. 
Br. 
Br. 
li.  Y. 
B.Y. 

(1.481 
11.489 
1.482 
1.482 
1.485 
1.482 
1.485 
1.487 
1.482 

l'.483 
1.483 
1.485) 
1.489J 

U85 
1.486 
1.487 
1.487 
1.488 
1.487 
1.496 
1.500 
1.500 
(1.487 
11.495 
1.487 
1.488 
/  1.488 
11.519 
[1.657 
11.665 
1.6543 
1.4833 
1.667 
1.488 
1.490 
1.507 
1.490 

(1.483 
(1.491 
(1.490 
(1.491 

(1.491 
(1.507 

T.492 
1.507 
1.5123 
1.4503 
1.4416 
1.542 
1.535 
1.494 
(1.494 
(1.497 

L495 
1.4985 
1.4929 
1.515 
1.500 
1.500 
1.500 
(1.500 
J  1.503 
1  1.505 

1.500 
1,504 
1.5133 
1.514 

Br. 
B.Y. 
Br. 
B.Y. 
Br. 
Br. 
B.Y. 
N. 
W. 
B.Y. 
Br. 

B.Y. 

Br. 
B.Y. 

B.Y. 

B.Y. 
Br. 
B.Y. 
B.Y. 
B.Y. 
Br. 
B.Y. 
W. 
B.Y. 
C. 
N. 
Br. 
B.Y. 
B.  V. 
Br. 
W. 
B. 
W. 
Br. 
M. 
M. 
N. 
Br. 
He. 
Br. 
Br. 
W. 
B.Y. 
B.Y. 
Br. 
B.Y. 
Br. 
B.Y. 
W. 
B.Y. 
Br. 
B.Y. 
Br. 
B.Y. 
B.Y. 
M. 
M. 
M. 
W. 
W. 
Br. 
B.Y. 
Br. 
W. 
Br. 
B.Y. 
He. 
He. 
Br. 
W. 
B.Y. 
Br. 
Br. 
B.Y. 
Br. 

W. 
W: 
He. 

Bos. 

Fresh  yolk  of  an  egg   
Sulphuric  acid  (S.  G.  1.7)    

1.428 
.      1.429 

Oil  of  lemon    

Ditto      ditto    (?  S.  G.)     

1.430 
1.435 
1.440 
(1.433 
(1.436 
[1.433 
\1.449 
1.437 

Carbonate  of  potash  (?)  

Oil  of  pennyroyal  

Ditto     

Linseed  oil  (S.  G.  0.932)  
Linseed  oil    

Ditto     

. 

(1.441 
(1.442 
[1.446 
11.454 
1.452 
.     1  453 

Spermaceti  (melted)  

Oil  of  wormwood      

O'l    f              °<1  "  " 

Ditto  

Bees  wax,  melted  

1.453 

1.457 
.     1.476 
1.457 
.     1.467 

Florence  oil  „  

Oilof  dill  seed    

.     1.475 

Oil  of  feugreek  (?  fenugreek)  .  . 

Alum                                                                .    . 

1.457 
1.458 
1.488 
1.460 
1.462 

1U83 
1.465 
{1.467 
1.467 
1.475 
(1.468 
1  1.473 
(1.469 
(1.472 
(1.470 
(1.473 
(1.4691 

(1481 
11.483 
1.470 
1.471 
1.475 
1.476 
1.476 
1.482 
1.485 
1.486 
1.47835 
11.467* 
1.469 
1.470 
1.4705 
.  1.476 
(1.471 
(1.473 
1.471 
(1.470 
(1.471 
1.473 

1.475 
1.475    1 

1.475 
1.476 
1.476 
1  477 

Ditto-  

Ditto  (SGI  714)                              .. 

f  

Ditto     

Tallow  (melted)                                                  . 

((S.G.  =  0996).. 

Sulphate  of  magnesia  (double  ?  least  refraction). 
Borax,  (S.  G.  —  1.714)   

PPP 

Ditto     ordinary  index  

Ditto    (S.G.  =  2.72)  
Sulphate  of  magnesia  (?  greatest  retraction)  .  .  .  . 

Ditto    

Tallow  (cold)    

Spirit  of  turpentine,  (S.  G.  0.874)  

Ditto     

Ditto     

Ditto         

Ditto  (common)   

Ditto  S  G  —  0585  (ray  E)  

Oil  of  Angelica    

Bees  wax,  cold    

Oil  of  bergamot  

Ditto  14°  Reaum  

Ditto  

'  p          '  s 

Palm  oil    

Naphtha 

Ditto              (mean  red)    

Ditto               (tartrate  of  potash  and  soda)  

Treacle  

Oil  of  dill  seed 

Yolk  of  an  egg  (dry)    

1.477 

1.479 
1.479 
1.481 

Oil  of  beech  nut   

Oil  ol  cajeput       

Glass,  plate  and  crown,  various  specimens  : 

Ditto  French  plate  

•  The  S   G.  ftf  Newton'i  specimen  was  0.913. 

Ditto  English  plate  (extreme  red)  
Ditto  plate   . 

Part  TV. 


LIGHT. 


571 


Light. 


1.517       W. 
1.525       W. 
1.526       Bos. 
1.526       He. 
1.527       Br. 
1.529       Bos. 
1.5301     He. 

1.5314     Fr. 
1.532       C. 

1.5330     Fr. 

1  1  '^fi  (    ^' 
L534       Br. 
1.538       Bos. 
1.542       Bos. 
1.543       W. 
1.544       Br. 
1.545       W. 
1.550       N. 
1.5631     Fr. 
1.573       C. 
1.582      Br. 
pecimens  of  this 
ead. 
(1.503       B.Y. 
1  1.535      W. 
{1.503       B.Y. 
1.507       Br. 
1.510       B.Y. 
1.504      B.Y. 
1.504       B.Y. 
1.613      B  Y. 
(1.505       W. 
11.507       B.Y. 
1.506       B.Y. 
(1.506       Br. 
11.507       B.Y. 
f  1.507       W. 

•T'514l    R  V 
(1.51G/    B'Y- 

M.528       Br. 
1.508      Br. 
/1.  508       Br. 
11.578       B.Y. 
1.510       B.Y. 
f  1-5121 
11.526/     B'Y- 
1.512       Br. 
1-513       B.Y. 
1.514       W. 
(1.5141 
U.517/    W- 
1.514       Br. 
1.335       Br. 
1.524       C. 
1.524       N. 
1.515       N. 
1.5153     Br. 
1.516       Br. 

1517       Br! 
(1.517       B.Y. 
1  1.524       Br. 
1.518       Br. 
(1.529       Br. 
11.575       Br. 
(1.520       Br. 
(1.66         W. 
1.522       Br. 
(1.52+     W. 
11.525       Br. 
U.528      B.Y. 
(1.522       B.Y. 
J  1.532      Br. 
1  1.536       W 
U.544       Eul. 
f  1.524       W. 
{  1  .534       Br. 
11.557       B.Y. 

1.525       W. 
1.536      Br. 
1.488       N. 
1.527       Br. 
1.527       Br. 
,  1.528       W. 
\  1.532      B.Y. 
11.549       Br. 
1.529       B.Y. 
1.530       W. 
/•1.531       W. 
1.581       B.Y. 
\  1.586       Br. 
[1.588       B.Y. 
1.531       Br. 
1.552       Br. 
(1.532       B.Y. 
11.544       Br. 
1.532       C. 
1.532       Br. 
1.544      Br. 
1.532       Br. 
1.533       B.Y. 
f  1.5471         Y 
{  1.565  /    B'Y- 
1.5348     M. 
1.6931     M. 
[1.535       W. 
\  1.547      Br. 
U.550       B.Y. 
(1.535       W. 
\  1.539       B.Y. 
U.560       Br. 
1.535       W. 
(1.535       W. 
1  1.546      B.Y. 
(1.535       W. 
J  1.549       Br. 
1  1.553       B.Y. 
(1.535       W. 
11.539       B.Y. 
1.535      W. 
1.541       B.Y. 
1.545       B.Y. 
1.555       Br. 
1.536       Br. 
1.536       B.Y. 
(1.536      B.Y. 
1  1.601       Br. 
1.538       Br. 
1.556      Br. 
(1.538       Br. 
1  1.541       B.Y. 
1.540       Br. 
1.542      W. 
1.543      W. 

1.5431     He. 
1.543       Br. 
1.700       Br. 
1.544      Br. 
1.544       B.Y. 
1.545      N. 
1.557      Br. 
(1.545       B.Y. 
11.557       Br. 
1.545      B.Y. 
1.545       B.Y. 
(1.546       B.Y. 
1  1.560       Br. 
f  1.546       B.Y. 
11.554       Br. 
1.547      Br. 
1.547       W. 
1.5484    M. 
1.5582    M. 
1.562       W. 
1.562       Br. 
1.563       N. 
(1.5681 
11.575J    C' 

"  A  selenites,  S.  G.  2.252"  

Ditto  crown,  a  prism  by  Dollond,  (extra  red)    .  . 

Citric  acid  

Canada  balsam  

Ditto  crown,  another  prism  by  Dollond,  (extra  red) 
Ditto  Fraunhofer's    crown,    No.     13,    (ray   E.) 

Balsam  of  Gilead  

Ditto  yellow  plate   S.  G.  2  52  

Crystalline  of  ox  (dried)  and  of  a  fish 

Ditto  Fraunhofer's    crown,    No.    9,     (ray    K,) 
S.  G.2535    

Pitch 

Ditto  plate         

Ditto  ditto    

Ditto  St.  Gobin   

Brazil  pebble,  (S.  G.  2.62)  

Ditto  old  plate  

Glass  of  phosphorus  (fused  phosphoric  acid).  .  .  . 
Solid  phosphoric  acid       

Ditto  Fraunhofer's  crown,  M,  S.  G.  2.756,  (ray  E) 
Ditto  plate,  (S.  G.  2  76)  

Glass  of  borax  (fused  borax)  

Manna  

N.  B.  It  is  probable  that  the  more  refractive  s 
list  are  low  flint  glasses,  containing 

Arragonite,  extraordinary  index  

Ditto,  ordinary     

Elemi    

Mastic  

Starch  (drv)  

Araeniate  of  potash      .    . 

Ditto,  two  days  exposed  

Birdlime    

Copal    

Oil  of  cloves  

Stilbite 

Ditto      

Ditto  (melted)       
Ditto  (after  melting)    

Felspar     

Oil  of  mace  

Gum  Arabic  
Ditto  (not  quite  dry) 

Mellite  

Nitre,  greatest  index    
Ditto,  least   

Box-wood     .  . 

Ditto     
"Niter"  (?)  S.  G.  1.9  

Colophony     

Apophyllite,  the  variety  which  exhibits  white  and 

Dantzic  vitriol  (sulphate  of  iron)     
Nadelstein  from  Faroe  
Mesotype,  least  index   

Carbonate  of  strontia,  least  refraction  
Dichroite  (iolite)  

Sulphate  of  zinc,  ordinary  refraction 
Myrrh    . 

Petroleum  
Sal  gemma;,  S.  G.  2.143  (rock  salt)    
Ditto  (rock  salt)   

Tartaric  acid,  least  refraction  . 

Chio  turpentine     

Gum  sagapenum  

Gum  dragon  (Tragacanth)  

Glass  of  borax  1,  silex  2  

Gum  lac,  or  Shell  lac  

Quartz,  ordinary  refractive  index 

Caoutchouc  .       . 

Rock  crystal  (double)  

Crystal  of  the  rock  (S.  G.  2.65) 

Rock  crystal  

Part  IV. 


4  E  2 


572 


LIGHT. 


Light. 


Amber 

Ditto,  (S.  G.  1.04) 


Resin 


Guiacum   ....... 

Glue,  nearly  hard 

Chalcedony 

Comptonite 

n  . 

°P'um 


1 


1.547 
1.556 
1.548 
l.552 
1559 
1.550 
1.553 
1.553 
1.553 
1.559 


Hyposulph  Ate  of  lime  (mean  red) 1.5611 

Ditto,  mean  yellow  green     • 1 .566 

Dragon's  blood 1 .562 

(1.565 
Horn  \lrs 

Pink,  coloured  glass  1 .570 

Assafcetida  1.575 

1 1.576 
Flint  glass  (various  specimens)  si. 578 

U.583 

Ditto,  a  prism  by  Dollond  (extreme  red)  1.584 

Ditto,  (extreme  red)  1.585 

Ditto,  another  specimen  1 .586 

Ditto 1.590 

Ditto 1.593 

Ditto 1.594 

Ditto 1 .596 

Ditto,  a  prism  by  Dollond  (extreme  red) 1.601 

Ditto  ditto,  marked  "  heavy,"  (extreme  red)  ....  1.602 

Ditto,  another  specimen    , 

Ditto 1.605 

Ditto,  Fraunhofer's  No.  3  (ray  E) 1.6145 

Ditto,  another  variety 1.616 

Ditto        ditto  1.625 

Ditto,  Fraunhofer's  No.  30  (ray  E) 1.6374 

Ditto        ditto          No.  23  (ray  E) 1 .6405 

Ditto        ditto         No.  13  (ray  E) 1.6420 

Anhydrite,  ordinary  index    1.5772 

Ditto,  extraordinary     1.6219 

f 1.578 
Gum  ammoniac j  j  ,-,92 

Hyposulphite  of  lime,  least  refraction   1.583 

Ditto,  greatest 1.628 

Balsam  of  styrax 1 .584 

Emerald    1.585 

1.5861 

Benzom to  1.596  / 

\      1.589 

Oil  of  cinnamon    \      1 .604  ) 

(to  1.632J 

Tortoise  shell      1.591 

'  1.593 

BalsamofPeru J1.597 

(1.605 
.1.596 

Guiacum   J  1.600 

(1.619 

Beryl      1.59M 

fl.60— 
I  I  610 
|1.627 
(1.628 

Ruby  red  glass 1 .601 

Essential  oil  of  bitter  almonds 1.603 

Meionite   1 .606 

Purple  coloured  glass   1.608 

Resin  of  jalap 1 .608 

Hyposulphite  of  strontia,  least  refraction    1.60H 

Ditto  ditto  greatest          1.651 

Colourless  topaz      1.6102 

Bluish  topaz  (cairngorm)      1  624 

Brazilian  topaz,  ordinary  index   1.6325 

Ditto         ditto,  extraordinary 1 .6401 

Blue  topai,  Aberdeen 1.636 

Yellowtopaz 1.638 

Red  topaz 1 .652 

Green  coloured  glass    1.615 

„    .  (1.620 

Castnr 11626 

Sulphate  of  barytes,  ordinary      1.6-jOl 

Ditto,  extraordinary      1 .6352 


Balsam  of  Tolu. 


W. 

N. 

B.Y. 

B.Y. 

Br. 

B.Y. 

B.Y. 

Br. 

Br. 

B.Y. 

W. 

He. 

He. 

B.Y. 

Br. 

W. 

Br. 

B.Y. 

Br. 

He. 

W. 

He. 

He. 

W. 

Bos. 

Bos. 

Bos. 

Br. 

He. 

He. 

Br. 

Bos. 

M. 

Fr. 

Br. 

Bos. 

Fr. 

Fr. 

Fr. 

Bi. 

Bi. 

B.Y. 

Br. 

He. 

He. 

Br. 

Br. 

W. 

B.Y. 

B.Y. 

Br. 

B.Y. 

Br. 

B.Y. 

W. 

B.Y. 

Br. 

Br. 

W. 

B.Y. 

BY. 

Br. 

Br. 

lir. 

Br. 

Br. 

B  Y. 

He. 

He. 

Bi. 

Br. 

Bi. 

Bi. 

Br. 

Br. 

Br. 

Br. 

B.Y. 

Br. 

Bi. 

M. 


Sulphate  of  barytes 1.6468 

Ditto  ditto       ordinary  refraction  (along  the 

axis)  for  yellow  green  rays    1.6460 

Ditto,  another  specimen,  ditto,  red  rays      1.6459 

Ditto             ditto         for  yellow  green    1.6491 

A  "  psetido   topazius"   (S.  G.  4.27)   sulphate  of 

baryta    1.643 

Sulphate  of  barytes 1 .646 

Ditto            ditto      double,  greater  refraction. ...  1.664 

(1.624 

Oil  of  cassia     <  1 .631 

M.641 

Muriate  of  ammonia     1 .625 

Aloes     1.634 

Opal  coloured  glass 1.635 

Euclase,  ordinary  index  1.6429 

Ditto,  extraordinary 1 .6630 

Sulphate  of  strontia 1 .644 

Hyacinth  red  glass   1.647 

Mother  of  pearl    1.653 

Spargelstein 1 .657 

Epidote,  least  refraction 1.661 

Ditto,  greatest 1.703 

Tourmaline    1 .668 

Cryolite,  least  refraction 1.668 

Ditto,  greatest 1.685 

Chloruret  of  sulphur 1.67 — 

Nitrate  of  bismuth,  least  refraction,  about 1.67 — 

Ditto,  greatest,  about    1.89 — 

Sulphuret  of  carbon 1.678 

Orange  coloured  glass      1.695 

Boracite     1.701 

Glass  tinged  red  with  gold   1.715 

Glass,  lead  1,  flint  2    1.724 

Deep  red  glass 1 .729 

Nitrate  of  silver,  least  refraction      1.729 

Ditto,  greatest   1.788 

Glass,  lead  3,  flint  4 1.732 

Hyposulphite  of  soda  and  silver,  least  refraction  .  1.735 

Ditto,  greatest  1.785 

Axinite 1.735 

Nitrate  of  lead 1.758 

Cinnamon  stone   1.759 

Chrysoberyl 1 .760 

(1.756 

Spinelle    ^1.761 

U.812 

Felspar     1.764 

Sapphire,  (white)      1.768 

Ditto,         (blue) 1.794 

Hnbdlite |};™j 

Ruby     1.779 

Jargon  (orange  coloured) 1.782 

Glass,  lead  1,  flint  1  (Zeiher) 1.787 

Pyrope 1.792 

Labrador  hornblende    1.804- 

iMuriute  of  antimony  (variable)  about    1.8 

Arsenic 1 .81 1 

Carbonate  of  lead,  least  refraction 1.813 

Ditto  ditto       greatest         2.084 

Borate  of  lead,  fused  and  cooled  (extreme  red).  .  1.866 

Sulphate  of  lead   1.925 

Glass,  lead  2,  sand  1     1.987 

Zircon    1 .95 

Ditto,  least  refraction   1.961 

Ditto,  greatest  2.015 

Sulphur  (Hauy)        1.958 

Ditto     2.008 

Ditto      2.04 

Ditto,  native     2.115 

Ditto,  melted .......  2.148 

Calomel     1.970 

Tungstate  of  lime,  least  refraction 1 .970 

Ditto,  greatest   2.129 

Glass  of  antimony  12216 

Glass,  lead  3.  flint  1  (by  Zeiher)  2.028 

Kruly  oxide  of  iron  2.1  — 

Silicate  of  lead,  atom  to  atom,  extreme  red 2.123 

(2.125 
Phosphorus  J2.224 

\2.260 


M. 

He. 
He. 
He. 

N. 

W. 

Br. 

B.Y. 

B.Y. 

Br. 

Br. 

B.Y. 

Br. 

Bi. 

Bi. 

Br. 

Br. 

Br. 

Br. 

Br. 

Br. 

Br. 

Br. 

Br. 

He. 

He. 

He. 

Br. 

Br. 

Br. 

Br. 

Br. 

Br. 

Br. 

Br. 

He. 

He. 

Br. 

Br. 

Br. 

Br. 

He. 

Br. 

W. 

Br. 

W. 

Br. 

He. 

Br. 

Br. 

Br. 

Zei. 

Br. 

He. 
W. 
W. 
Br. 
Br. 

He. 

Br. 

W. 

W. 

Br. 

Br. 

Ha. 

B.Y. 

W. 

Br. 

Br. 

Br. 

Br. 

Br. 

W. 

Br. 

Z. 

Y. 

He. 

B.Y 

Br. 

Br. 


Part  IV. 


LIGHT. 


573 


Light.          Nitrite  of  lead  (biaxal,  ?  quadro-nitrite)  in  six- 

_          _  .  sided  prisms,  ordinary  refraction 2.322 

Diamond  (S.  G.  =  3.4)   2.439 

Ditto 2.470 

Ditto  (brown  coloured)    2.4S7 

Ditto  (examined  by  Rochon) 2  755 

(from  204 

Plumbago    tto...  2.44 


He. 

N. 


Chrcmate  of  lead 


(2.479 
least  refraction  ............    -2.500 

1  2.503 


Br. 
Br. 
Br. 
Br. 
Br. 
Br. 
Br. 
Br 


Pirt  I 


Ro.         Octohedrite   .............................  2.500 

W.  Realgar,  artificial  ..........................  2.549 

W.          Red  silver  ore  ............................  2.564 

Mercury  (probable,  see  Art.  594)  ............  5.829 

In  casting  our  eyes  down  the  foregoing  Table,  we  cannot   but  be  struck  with  the  looseness   and  vagueness       1117 
of  those  results  which  refer  to  bodies  whose  chemical  nature  is  in  any  respect  indeterminate.     The  refractive  Remarks  on 
indices  assigned  to  the  different  oils,  acids,  &c.  though  no  doubt  accurately  determined  for  the  particular  specimens  the  Table 
under  examination,  are  yet,  as  scientific  data,  deprived  of  most  of  their  interest  from  the  impossibility  of  stating  ofRefractive 
precisely  what  was  the  substance  examined.     Most  of  the  fixed  oils  are  probably  (as  appears  from  the  researches  -n"'ces-- 
of  Chevreul)   compounds,   in  very  variable  proportions  of  two   distinct  substances,  a  solid,  concrete   matter, 
(stearine,)  and  a  liquid,  (elaine,)  and  it  is  presumeable,  that  no   two  specimens  of  the  same  oil  agree  in   the 
proportions.     This  is,  probably,  peculiarly  the  case  with  the  oil  of  anise  seed,  which  congeals  almost  entirely 
with  a  very  moderate  degree  of  cold..    An  accurate  reexamination  of  the  refractive  and  dispersive   powers  of 
natural   bodies    of   strictly   determinate   chemical    composition,    and    identifiable  nature,    though  doubtless   a 
task  of  great  labour  and  extent,  would  be  a  most  valuable  present  to  optical  science.     Fraunhofer's  researches 
have  shown  to  what  a  degree  of  refinement  the  subject  may  be  carried,  as  well  as  the  important  practical  uses 
to  which  it  may  be  applied.     The  high  refractive  power  of  oil  of  cassia,   accompanied  by  a  corresponding 
dispersion,  has  led  Dr.  Brewster  to   conceive  the  existence  in  it  of  some  peculiar  chemical  element  not  yet 
cognisable  by  analysis.     The  low  refractions  of  the  oils  of  box-wood  and  ambergris  are  not  less  remarkable. 
It  is  among  the  artificial  salts,  however,  that  the  widest  field  is  open  for  the  application  of  precise  research,  and 
one  in  which  a  rich  harvest  of  important  results  would,  in  all  probability,  amply  repay  the  trouble  of  the  inves- 
tigation, whether  considered  in  an  optical,  a  chemical,  or  a  crystallographical  point  of  view. 


The  fraction  P  = 


'-  1 


where  fi  is  the  refractive  index,  and  s  the  specific  gravity  of  the  medium,  expresses      1118. 

Table  of 


(in  the  doctrine  of  emission)  the  intrinsic  refractive  energy  of  its  molecules,  supposing  the.  ultimate  atoms  of  all  intrtn'sic 
bodies  equally  heavy.     The  following  results  have    been  stated  by  various  authors,  as  its  values  for  bodies  most  Refractive 
widely  differing  in  their  chemical  and  mechanical  relations.  Power* 

I.     Gases,  taking  the.  value  of  P  for  atmospheric  air  as  unity.     (From  Biot's  Precis  Elementaire,  ii.  224.) 


Oxygen     0.86]  61 

Air    1.00000 

Carbonic  acid  .  1.00476 


Azote   1.03408 

Muriatic  gas    1.19625 

Supercarburetted  hydrogen  1.81860 


Carburetted  hydrogen. .    2.09270 

Ammonia "Z.ie-'Sl 

Hydrogen    6.61436 


II.    Direct  values  of  P  given  by  the  formula. 

Those  marked  Dulong  are  computed  from  the  refractive  indices  of  Dulcmg  in  the  last  table. 


Brewster. 

Brewster. 

Brewster. 

Dulong. 

Dulong. 

Newton. 


Tabasheer    0.0976 

Cryolite 0.2742 

Fluorspar 0.3426 

Oxygen   0.3799 

(0.3629 
Sulphate  of  barytes    . .  I  „  3979 

Sulphurous  acid  gas  . .  0.44548  Dulong. 

Nitrous  gas 0.44911  Dulong. 

(0.4528  Dulong. 

Air \  0.4530  Biot. 

(0.5208  Newton. 

Carbonic  acid 0.45372  Dulong. 

Azote 0.4734  Dulong, 

Chlorine 0.48133  Dulong. 

Glass  of  antimony  ....  0.4864  Newton. 

Nitrous  oxide 0.5078  Dulong. 

Phosgen  0.5188  Dulong. 

Selenite  0.5386  Newton. 

Carbonic  oxide  0.5387  Dulong. 

Quartz  0.5415  Malus. 

f  0.5450     Newton. 

Rock  crystal   <  „  ..„, 

( 0.6536     Brewster. 

Vulgar  glass    0.5436     Newton. 


Muriatic  acid  glass  ....    0.5514  Dulong. 

Sulphuric  acid    0.6124  Newton. 

r  ,  1 0.6424  Malus. 

Calcareous  spar <  -  e-o^  M 

( 0.6.136  Newton. 

Sal  gem   0.6477  Newton. 

•Muriate  of  soda 1  .'2086  Brewster. 

Alum    0.6570  Newton. 

Nitric  acid O.G67G  Brewster. 

Borax O.fi716  Newton. 

Niter    0.7079  Newton. 

•Nitre 1.19'62  Brewster. 

Hydrocyanic  acid    ....    0.7366  Dulong. 

Ruby 0.7389  Brewster. 

Dantzic  vitriul,(sul.  iron)  0.7551  Newton. 

Muriatic  ether  (vapour)     0.7552  Dulong. 

Brazilian  topaz    0.7586  Brewster. 

Rain  water 0.7845  Newton. 

Flint  glass  (mean) 0.7986  Brewster. 

Cyanogen     0.8021  Dulong. 

Sulphuretted  hydrogen  .    0.8419  Dulcng. 

Gum  Arabic 0.8574  Newton. 

Vapour  of  sulphuret  of 

carbon 0.8743 


Vapour  of  sulph.  ether. .    0.9 1 38 
Protophosphuretted  hydr.  0.9680 

Ammonia 1.0032 

Rectified  spirits  of  wine    1.0121 
Carbonate  of  potash   ..    1.0227 

Cl.romateof  lead   1.0436 

Olefiantgas 1.0654 

*Muriate  of  ammonia..    1.1-90 
Carburetted  hydrogen       1.2204 

Camphor 1..2551 

Olive  oil 1.2607 

Linseed  oil 1.2819 

Beeswax    1.3308 

Spirit  of  turpentine    ..    1.3222 

Amber     1  3654 

Octohedrite 1.3816 

Diamond     1.4566 

Realgar 1.6666 

Ambergris    1.7000 

Mercury  (probable).  .  .  .    2.4247 

Sulphur    '2.2000 

Phosphorus 28857 

Hydrogen    3  0953 


Dulong. 

Dulonu. 

Dulong. 

Dulong. 

Brewster. 

Brewster. 

Dulong. 

Brewster. 

Dulong. 

Newton. 

Newton. 

Newton. 

Malus. 

Newton. 

Newton. 

Brewster. 

Newton. 

Brewster. 

Brewster. 

Brewsler. 
Brewster. 
Dulong. 


Dulong. 

The  results  marked  with  an  asterisk  in  this  table  have  probably  originated  in  some   miscalculation.     As      1119. 
hydrogen  stands  highest  in  this  scale,  so  it  is  probable  that  fluorine,  should  we  ever  obtain  it  in  an  insulated  Remarks  OP 
state,  would  prove  the  lowest.     The  optical  properties  of  tabasheer,  in  all  points  of  view,  are  strange  anomalies,  'his  Table. 

ul  —  1 

It  will  be  observed,  that  the  function only  expresses  the  intrinsic  refractive  power  on  the  hypothesis  of  the 

s 

infinite  divisibility  of  matter,  and  the  equal  gravitating  power  of  every  infinitesimal  molecule.  But  if,  as  modern 
Chemistry  indicates,  material  bodies  consist  of  a  finite  number  of  atoms,  differing  in  their  actual  weight  for  every  dif- 
ferently compounded  substance,  the  intrinsic  refractive  energy  of  the  atoms  of  any  given  medium  will  be  the  product, 
of  the  above  function  by  the  atomic  weight.  This  will  alter  totally  (he  order  of  media  from  what  obtains  in  the 
foregoing  table.  Thus,  the  weight  of  the  atom  of  hjdrogen  being  the  least,  and  that  of  mercury  one  among  the 


574 


LIGHT. 


Light. 


1120. 

Table  of 

Dispersive 

Powers. 


greatest  in  the  chemical  scale,  such  multiplication  will  depress  the  rank  of  the  former,  and  exalt  that  of  the  latter,     Part  IV. 
so  as  to  separate  them  entirely  from  the  proximity  they  now  hold.     A  distinction,  too,  will  require  to  be  regarded  ^^-v^. 
between   compound  and  simple  atoms.     But  as  these  considerations  are  peculiar  to  the  system  of  emission,  we 
shall  not  prosecute  them  farther  in  detail 

The  dispersive  powers  of  bodies  afford  another  very  interesting  and  distinctive  chaiacter.  Of  these,  Dr. 
Brewster,  in  his  Treatise  on  New  Philosophical  Instruments,  has  given  the  following  extensive  table,  almost 
entirely  from  his  own  observation. 

TABLE  OF  DISPERSIVE  POWERS. 


Column  1  contains  the  name  of  the  medium  ;  column  2  the  value  of  the  function 

r 
fi  $  simply,  &  ft  being  the  difference  of  refractive  indices  of  extreme  red  and  violet  rays. 


• ;  column  3,  that   of 


** 

Au- 

^ 

Au- 

Dispersive Powers. 

?-.  i 

P- 

thor. 

/•-i 

/*• 

thor. 

Chrom  lead  greatest  estimated 

0.400 

0.770 

R 

Oil  brick  . 

0  046 

0021 

B 

Ditto              greatest  exceeds 
Realgar,  melted,  different  kind 

0.296 
0.267 

0.570 
0.394 

B. 
R 

Flint  glass,  (Boscov.  lowest) 
Nitric  acid  

0.0457 
0.045 

0.019 

B. 
B. 

Chrom   lead   least  refraction  . 

0.262 

0.388 

R 

Oil  lavender   

0  045 

0  021 

B. 

0  255 

0  374 

R 

Balsam  of  sulphur 

0  045 

0  023 

B. 

Oil  cassia              

0.139 

0.089 

R 

Tortoise  shell  

0  045 

0  027 

B. 

0.130 

0.149 

R 

Horn  

0.045 

0.025 

R 

Phosphorus               

0  128 

0.156 

R 

Canada  balsam  

0  045 

0  024 

B 

Balsam  Tolu         

0.103 

0.065 

R 

Oil  marjorum  

0.045 

0.022 

B. 

0  093 

0.058 

R 

0  045 

0  024 

B. 

Carb   lead  greatest   

+  0.091 

+  0.091 

R 

Nitrous  acid  (?)    

0.044 

0.018 

B. 

0.085 

0.058 

R 

Cajeput  oil  

0.044 

0.021 

B. 

Oil  aniseed                        ...    . 

0.074 

0.044 

R 

Oil  hyssop    

0.044 

0.022 

B. 

Balsam  styrax      

0.069 

0.039 

R 

Oil  rhodium   

0.044. 

0.022 

B. 

Ouiacum     

0.066 

0.041 

R. 

Pink  coloured  glass  

0.044 

0.025 

R 

Carb   lead   least  refraction   .  . 

0.066 

0.056 

R 

0.044 

0.021 

R 

Oil  cummin    

0.065 

0.033 

R. 

Oil  POPDV   . 

0.044 

0.020 

R 

0.063 

0.037 

R. 

Jargon,  greatest  refraction    .  . 

0.044 

0.045 

R 

0  062 

0.032 

R 

0  043 

0  016 

B. 

0  062 

0.033 

R 

0.043 

0.024 

B 

Green  glass         

0.061 

0.037 

R 

Nut  oil    i    

0.043 

0.022 

R 

0  060 

0.056 

R 

Burgundy  pitch   

0.043 

0.024 

B. 

Deep  red  glass     

0.060 

0.044 

R 

0.042 

0.020 

R 

0  060 

0.032 

R 

Oil  rosemary     

0.042 

0.020 

R 

0.060 

0.038 

R 

Felspar  

0.042 

0.022 

R 

0  057 

0.032 

R 

Glue    

0.041 

0.022 

R 

Oil  sweet  fennel  seed    

0.055 

0.028 

R 

Balsam  capivi     

0.041 

0.021 

R 

0.054 

0.026 

R 

0.041 

0.021 

R 

0.053 

0.042 

R 

Stilbite    

0.041 

0.021 

R 

0.053 

0.029 

R 

0.041 

0.023 

B. 

Flint  glass  (Boscov  greatest) 

0  0527 

Bos. 

0  040 

0.019 

B. 

0.052 

0.028 

R 

Spinelle           

0.040 

0.031 

B. 

Oil  pimento    

0.052 

0.026 

R 

Carb.  lime,  greatest  refraction 

0.040 

0.027 

R. 

0  052 

0.032 

R 

Oil  rape  seed  

0.040 

0.019 

B. 

0.051 

0  031 

B. 

0.040 

0  023 

B 

0  051 

0.025 

R 

Gum  elemi   

0.039 

0.021 

B. 

0.050 

0.024 

R 

Sul.  iron  

0.039 

0.019 

R 

Oil  fen  (^fenu)  °reek 

0  050 

0.024 

B 

Diamond    

0.038 

0.056 

H. 

0.049 

0.022 

R 

0.038 

0.018 

R 

0.049 

0.024 

R 

0.038 

0.022 

B. 

0.049 

0.024 

R 

White  of  egg  . 

0.037 

0.013 

B 

Oil  dill  seed 

0.049 

0.023 

R 

0.037 

0.016 

R 

0.049 

0.023 

R 

Gum  myrrh  

0.037 

0.020 

R 

0  048 

0.029 

R 

Beryl  

0.037 

0.022 

R 

Chio  turpentine        

0.048 

0.028 

R 

Obsidian  

0.037 

0.018 

R. 

0  048 

0.028 

R 

Ether          

0.037 

0.012 

R, 

0  048 

0  028 

B 

Selenite             

0.037 

0.020 

R 

Oil  lemon                     . 

0.048 

0.023 

R 

0.036 

0.017 

R, 

0  047 

0.022 

R 

0.036 

0.018 

R 

Oil  chamomile   

0  046 

0.021 

R 

Sulphur  copper  

0.036 

0.019 

R. 

0.046 

0.025 

R 

Crown  glass,  very  green    .... 

0.036 

0.020 

R 

Carb   strontia   greatest  refrac 

0.046 

0  032 

B 

Gum  Arabic  

0.036 

0.018 

R 

LIGHT. 


575 


Dispersive  Powers, 

3^ 

lf*. 

Au- 
thor. 

Dispersive  Powers. 

HP 

5^ 

Au- 
thor. 

^-1 

p.-  \ 

Sugar,  cooled  from  fusion     .  . 
Jelly  fish  (medusa  aquora)  body 
Water 

0.036 
0.035 
0.035 
0.035 
0.035 
0.035 
0.035 
0.035 
0.035 

0.0346 
0.033 

0.033 
0.033 
0.033 
0.033 
0.032 
0.032 
0.032 
0.032 

0.020 
0.013 
0.012 
0.012 
0.012 
0.019 
0.027 
0.018 
0.024 

0.027 

0.026 
0.022 
0.018 
0.012 
0.012 
0.017 
0.017 

B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 

Bos. 
B. 

Bos. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 

0.031 
0.031 
0.030 
0.030 
0.030 
0.029 
0.029 
0.028 

0.027 
0.027 
0.026 
0.026 
0.026 
0.026 
0.025 
0.025 
0.024 
0.024 
0.022 
0.022 

0.014 
0.017 
0.016 
0.014 
0.022 
0.011 
0.019 
0.019 

0.015 
0.014 
0.015 
0.016 
0.021 
0.016 
0.019 
0.025 
0.015 
0.010 
0.007 

B. 
He. 
B. 
B. 
B. 
B. 
B. 
B. 

Rob. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 
B. 

Apophyllite  (leucocyclite)  
Tartaric  acid     

Aqueous  humour,  haddock  eye 
Vitreous  humour,  haddock  eye 

Alcohol  

Rubellite 

Sulph.  barytes    

Tourmaline  

Epidote 
Common    glass,     Boscovich's 

Crown    glass,    Leith,     (Robi- 
son,)  cited  by  Brewster.  .  .  . 
Carb.  strontia,  least  refraction 
Rock  crystal   

Common     glass,     Boscovich  s 
lowest,  cited  by  Brewster  .  . 

Carb.  lime,  least  refraction    .  . 
Blue  sapphire     

Chrysolite   

Bluish  topaz,  cairngorm    .... 
Chrysoberyl    

Blue  topaz,  Aberdeenshire    .  . 
Sulph.  strontia  

Phosphoric  acid,  solid  prism  . 
Plate  irlass  .  . 

Fluor  spar    .          

Crvolite  .  . 

Part  IV. 


Respecting  the  results  in  this  table,  the  remark  applied  to  that  of  refractive  indices  may  be  yet  more  strongly      1121. 
urged.     The  whole   stands  in  need   of  a  radical   reinvestigation.     Those  only,  however,  who  have  had   some  Remark  on 
experience  of  the  difficulties  in  the  way  of  a  strict  scientific   examination  of  dispersive  powers,  can  appreciate  tlle  TaWe  o( 
either  the  labour  of  such  a  task,  or  the  merit  of  Dr.  Brewster  in  his  researches,  which  we  must  not  be  understood  ?'*pee(rs've 
as  in  the  slightest  degree  depreciating  by  this  remark.     But  the  refinements  of  modern   science  are  every  day 
carrying  us  beyond  all  that  could  be  contemplated  in  its  earlier  stages,  and  it  is  matter  of  congratulation,  rather 
than  disappointment,  to  every  true  philosopher,  to  see  his  methods  replaced  by  others  more  powerful,  and  his 
results  rendered  obsolete  by  the   more  exact  conclusions  of  his  successors.     What  is  now  chiefly  wanted  is  a 
knowledge  of  the  whole  series  of  refractive  indices  for  the  several  definite  rays  throughout  the  spectrum,  under 
uniform  circumstances,  and  for  all   media  whose  chemical  and  other  characters  are  sufficiently  definite  and  con- 
stant to  enable  us  to  identify  and  reproduce  them  in  the  same  state,  at  all  times.     The  researches  of  Fraunhofer 
and  Arago  have  shown  that  accuracy  in  the  determination  of  refractive  indices  sufficient  for  the  purpose,  may  be 
attained,  and  we  trust,  therefore,  that  this  great  desideratum  will  not  long  remain  unsupplied. 

To  the  substances  in  the  table  many  important  remarks  apply.  In  general,  high  refractive  is  accompanied  by  1122. 
high  dispersive  power  ;  but  exceptions  are  endless,  especially  among  the  precious  stones,  of  which  diamond 
affords  a  striking  instance.  Particular  bodies  seem  to  carry  their  dispersive  as  well  as  their  refractive  powers 
with  them  into  their  compounds,  and  that  more  evidently,  because  by  the  peculiar  mode  in  which  the  dispersion 
is  represented,  the  state  of  condensation  is  eliminated.  Thus,  fluorine,  and  even  oxygen,  appear  to  exercise  a 
very  lowering  influence  on  the  dispersive  powers  of  their  compounds,  while  hydrogen,  sulphur,  and  especially 
lead,  act  with  great  energy  in  the  opposite  sense.  The  contrast  between  the  oils  of  ambergris  and  cassia,  is  at  Experiment 
least  as  remarkable  in  point  of  dispersive  as  of  refractive  power.  The  following  experiment  would  seem  to  point  o"  °''  °f 
out  the  hydrogen  of  the  latter  oil,  as  the  principle  to  which  its  extraordinary  dispersion  is  due,  and  is  otherwise  cassia- 
instructive,  as  exemplifying  strongly  the  independence  of  the  two  powers  inter  se.  A  stream  of  chlorine  was 
passed  through  oil  of  cassia  till  it  refused  to  act  any  farther.  The  oil  was  at  first  'greatly  deepened  in  colour, 
but  as  the  action  proceeded,  it  changed  to  a  much  lighter  ruddy  j'ellow,  which  it  retained  till  the  action  was 
complete,  (and  which  in  a  few  days  changed  to  a  fine  rose  red.)  Copious  fumes  of  muriatic  acid  gas  were  given 
off  during  the  whole  process,  indicating  the  abstraction  of  abundance  of  hydrogen,  and  at  length  the  oil  was  con- 
verted into  a  viscous  mass,  drawing  out  into  long  threads,  having  entirely  lost  its  peculiar  perfume,  and  acquired  a 
pungent,  penetrating  scent,  and  an  acrid,  astringent  taste,  totally  unlike  its  former  aromatic  flavour.  It  was  inflam- 
mable, though  less  than  before,  burning  with  a  flame  green  at  the  edges,  indicating  the  presence  of  chlorine.  Its 
refractive  power  was  very  little  diminished.  A  drop  being  placed  in  the  angle  of  two  glass  plates,  and  close  to 
it  a  drop  of  unaltered  oil  of  cassia,  the  spectrum  of  a  line  of  light  was  viewed  at  once  with  the  same  eye  through 
both  the  media.  They  still  formed  a  continuous  line,  the  spectrum  of  the  unaltered  oil  being  more  refracted  by 
only  about  one-fourth  the  breadth  of  that  of  tlie  altered  specimen.  But  the  dispersive  power  of  the  latter  was 
most  remarkably  diminished,  being  brought  below  not  only  that  of  the  unaltered  oil,  but  even  below  that  of  flint 
glass.  When  the  dispersion  of  the  unaltered  oil  was  corrected  by  flint  glass,  that  of  the  altered  was  found  to  be 
much  more  than  corrected;  and  when  the  angle  of  the  glass  plates  was  such  that  the  dispersion  of  the  latter  was 
iust  corrected  by  a  prism  of  Dollond's  "  heavy"  flint,  whose  refracting  angle  =:  about  25°,  the  unconnected 
spectrum  of  the  former  was  about  equal  to  that  of  the  flint  prism.  The  dispersion,  then,  had  been  diminished 
to  half  its  former  amount,  while  the  refraction  had  suffered  hardly  any  appreciable  change.  (October  7,  1925.) 

The   angle  of  complete  polarization  of  a  ray  reflected  at  the  surface  of  a  medium,  affords  a  most  valuable 
character  in  mineralogy,  as  it  gives  at  once  an  approximation  to  the  refractive  index,  sufficient  in  a  great  variety 


1123 


576 


L  I  G  H  T. 


Light. 
v— s/— ^ 

L'se  of  the 

polarizing 
angle  as  a 
physical 
character. 

Action  of 
crystallized 
surfaces  on 
reflected 
light. 


1124. 

Table  of 
angles  be- 
tween the 
optic  axes 
of  crystals. 


of  cases  to  decide  between  two  substances,  which  might  be  otherwise  confounded  together,  and  inasmuch  as  it 
can  be  measured  on  any  single  surface  sufficiently  polished  to  give  a  regular  reflexion,  thus  enabling  us  to  apply 
this  character  to  minute  fragments,  or  to  specimens  set  as  jewels,  or  otherwise  too  precious  to  be  sacrificed ;  to 
opaque  bodies,  and  to  a  variety  of  other  cases  where  a  direct  measure  of  the  refraction  would  be  impracticable. 
It  has  not  escaped  the  acute  and  careful  observation  of  Dr.  Brewster,  that  the  polarizing  angle  on  the  surfaces  of 
crystallized  media  is  not  absolutely  the  same  in  all  planes  of  incidence  ;  and  the  deviation,  though  excessively 
small  when  the  natural  reflexion  is  used,  becomes  very  sensible,  and  even  enormous,  when  the  reflexion  is 
weakened  by  covering  the  surface  with  a  cement  of  a  refraction  approaching  that  of  the  medium,  so  as  to  allow 
only  those  rays  to  reach  the  eye  which  have  penetrated,  as  it  were,  to  some  minute  depth,  and  undergone  some 
part  of  the  action  of  the  crystal  as  such.  The  point  is  among  the  most  curious  and  interesting  in  the  doctrine 
of  reflexion,  and  we  regret  that  our  limits,  as  well  as  the  obscurity  still  hanging  over  it,  and  which  it  will 
require  much  elaborate  research  to  dissipate,  prevent  our  devoting  a  section  to  it,  but  we  must  be  content  to  refer 
the  reader  to  an  excellent  paper  on  the  subject  by  that  Philosopher,  Philosophical  Transactions,  1819. 

The  angles  included  between  the  optic  axes  of  biaxal  crystals  is  a  physical  character  of  the  first  rank,  both 
on  account  of  its  distinctness,  its  extent  of  range,  (indifferently  over  the  whole  quadrant,)  and  its  immediate  and 
intimate  connection  with  the  state  in  which  the  molecules  of  the  crystals  subsist,  and  what  may,  loosely  speaking, 
be  termed  their  structure.  It  is,  however,  a  character  by  no  means  easily  determined  :  both  axes  rarely  lying 
within  one  field  of  view,  capable  of  being  examined  through  natural  surfaces,  and  requiring,  in  almost  all  cases, 
the  production  of  artificial  sections  ;  at  least,  this  is  the  only  safe  way  for  observations  of  the  tints,  for  the 
angles  at  which,  in  a  thin  parallel  plate,  the  several  successive  orders  of  colours  are  produced  in  situations 
remote  from  the  axes,  are  for  the  most  part  far  too  vague  to  lead  to  any  accurate  conclusion  as  to  the  position  of 
these  lines  within  the  plate,  not  to  speak  of  the  sources  of  fallacy  highly  coloured,  or  dichroite,  crystals  obviously 
present.  With  these  considerations  before  us,  we  cannot  but  be  struck  with  surprise  and  admiration  at  the 
unwearied  assiduity,  which  could  produce,  almost  unassisted,  a  table  of  results  so  extensive  and  so  valuable  as 
the  following. 

Table  of  the  Inclinations  of  the  Optic  Axes  in  various  Crystals. 


Carbonate  of  lime,  (Iceland  spar.") 
Carbonate  of  lime  and  magnesia,  (bitter 

spar.) 

Carbonate  of  lime  and  iron,  (brown  spar.) 
Tourmaline. 
Rubellite. 

Zircon. 
Quartz. 
Oxide  of  Iron. 


I.  UNIAXAL  CRYSTALS.     Inclination  =  0. 

Negative  Class. 

Corundum.  Idocrase,  (Vesuvian.) 

Sapphire,  Wernerite. 

Ruby.  Mica  from  Kariat. 

Emerald.  Phosphate  of  lead. 

Beryl.  Phosphato-arseniate  of  1  ad. 

Apatite.  Hydrate  of  strontia. 

Positive  Class. 

Tung-state  of  zinc.  I     Apophyllite. 

Titanite.  I     Sulphate  of  potash  and  iron. 

Bor&cite.  I     Superacetate  of  copper  and  time. 

Unclassed. 

Oxysulphate  of  irok. 


Arseniate  of  potash. 
Muriate  of  lime. 
Muriate  of  strontia. 
Suliphosphate  of  potash. 
Sulphate  of  nickel  aad  ( 


Hydrate  of  m  ignesia. 
Ice. 


Hyposulphate  of  lime. 


II.   BIAXAL  CRYSTALS. 


Names  of  crystals. 

Character  of  th 
principal  axis 
according  to 
I)r  IJrewster's 
system. 

Inclination  o 
optic  axes. 

Names  of  crystals. 

Character  of  th 

principal  ,-txi, 
according  in 
Dr.  lirew.itr'., 

SVMCIIV. 

Inclin.vioti  f>! 
optii  I 

Sulphate  of  nickel,  certain  specimens  .  . 

-f 

+ 

3°     V 
5     15 

6     56 

5     20 
6       0 
7    24 
11     28 
13     IS 
14       0 
18     18 
19     24 
25       0 
2?     51 
28       7 
28     42 
30      0 
31       0 
32      0 
34      0 
37      0 
35       8 
37     24 
37    42 
37     40 
38    48 
40       0 
41     42 
42       4 
43     24 
44     28 
4-1     41 

Mica  

+ 
+ 

+ 
-I- 

H- 

+ 
+ 

+ 

+ 

+ 
+ 

4')°      0' 
45       0 
45       8 
46     49 
4U     42 
49    r50° 
"id3      0' 
50       0 
51      16 
5  1     22 
5.')     21) 
56       6 
60       0 
62     1« 
62     50 
63      0 
65       0 
67       0 
70       1 
70     25 
70    29 
71      20 
79       0 
80       0 
80    30 
81     48 
82       0 
84     19 
84     30 
87    56 
88     14 
90       0 
90      0 

Sulphate  of  magnesia  and  soda 

Brazilian  topaz  (Brewslcr  and  Biot) 

Talc 

Muriosulphate  of  magnesia  and  iron 
Sulphate  of  ammonia  and  magnesia.  . 

Prmsiate  of  potash  (?  Ferrocyanate)  .  . 

+ 
+ 
+ 

+ 
+ 

Jolite  

Mica,  various  specimens  examined  by] 
M*  Biot                  ] 

Acetate  of  lead  

+ 

+ 

+ 

Peridot                 

Crystallized  Cheltenham  salts  
Succinic  acid,  estimated  at  about 

•• 

Anhydrite  ("examined  by  Biot)  

LIGHT. 


577 


Light.  Among  crystals  with  one  axis,  Dr.  Brewster  has  enumerated  the  Idocrase,  or  Vesuvian,  and  correctly.     Had     Part  IV. 

•"V""''  he  noticed,  however,  in  the  specimens  examined  by  him  the  very  striking  inversion  of  the  tints  of  Newton's  v-— v-~— 
scale  exhibited  in  the  rinses  of  that  now  before  us,  he  would  doubtless  have  made  mention  of  it.     We  insert  here      1125. 
the  scale  of  colours  exhibited  by  a  plate  cut  from  the  specimen  in  question,  (a  fine  large  crystal,)  as  affording  Remarks. 
^      another  remarkable  case  in  addition  to  that  of  the  hyposulphate  of  lime,  and  the  several  varieties  of  uniaxal  '."vert^ 
apophyllite  already  mentioned,  of  such  inversion.  Vesu\°ia-  ' 

Table  of  the  tints  exhibited  by  a  plate  of  Vesuvian,  thickness  =  0.11035  inch,  cut  a  little  obliquely  to  a  perpen- 
dicular to  the  axis. 


Angle  of 
Incidence. 

Ordinary  Image. 

Extraordinary  Image. 

«= 

Angle  of  Refraction  j. 

_L    6(jO    _L' 

No  light  passed     

No  light  passed. 

+  66       0 

Brick  red    

Dull  pale  Teen. 

-f-  *54       0 

Orange  red      

Fine  blue  "Teen. 

-f  60       0 

Tolerable  orange  pink  

Fine  bluish  green. 

-f   52       0 

Pale  yellow  pink   

Pale  yellowish  green. 

+  47       0 
4-42       0 

Pink,  with  a  dash  of  purple.  . 
Pale  neutral  purple   

Pretty  bright  yellow. 
Good  yellow  

i 

95°    56' 

+  37       0 

Yellow  less  bright. 

+  30       0 
+   15       0 

Very  pale  yellowish  white.  .  .  . 
Yellowish  white  

Sombre  brownish  yellow. 
Very  sombre  yellow  brown  .  . 

^ 

+   10       0 

Almost  totally  extinct   

to 

6     31   i 

+     3       0 

Yellowish  white  

Very  sombre  purplish  brown 

j 

±00 

Duskv  brownish  vellow. 

9       0 

Bluish  white  

Rather  dull  yellow. 

-    12       0 

Dull  purplish  blue    

Bright  yellow     

i 

+     7     48 

-    16       0 

Ruddy  purple  '   

Pale  yellow. 

-    19       0 
-   22       0 

Pink,  verging  to  brick  red    .  . 
Yellowish  red    

Imperfect  green. 
Tolerable  bluish  green. 

-    26       0 
-    23       0 

Yellow,  inclining  to  orange  .  . 
Bright  yellow     

Rich  greenish  blue. 
Blue  purple. 

-   28     30 

Bright  yellow     

1 

+  18     10 

-   29       0 

Bright  yellow     .... 

Ruddy  purple. 

-   30       0 

Crimson. 

-   32       0 

Good  pink. 

-   35       0 

Greenish  blue    

Orange  pink. 

-   37     30 

Pale  yellow. 

-   38     30 

Neutral  purple  

Pale  yellow      .        

i 

+  24       0 

-   39     15 

Ruddy  purple 

Greenish  yellow 

-    41     30 

Good  pink  

Good  green. 

-    45       0 

Pink  yellow    

Fine  greenish  blue. 

-    47     20 

Yellowish  white       

Blue  purple. 

-   47     30 

Yellowish  white 

Neutral  purple                   .... 

•2 

+  28     48 

-    48       0 

Very  pale  green  

Ruddy  purple. 

-    49     30 

Fine  green     

Good  pink. 

-    53       0 

Fine  blue  green  

Orange  pink. 

-    54       0 

Greenish  blue     

Yellow. 

-    54     — 

No  light  passed  

No  light  passed. 

The  first  ring,  it  will  be  observed,  in  calculating  from  this  table,  is  contracted  beyond  what  is  due  to  the  law 
of  the  sines,  probably  from  the  section  examined  not  passing  precisely  over  their  common  centre,  and  gives  a 
polarizing  power  greater  than  that  deduced  from  the  angles  corresponding  to  n  =  1,  n  =  -j,  n  =  2,  all  which 
agree  in  assigning  41.35  nearly  as  the  measure  of  the  power  in  question.  See  Art.1126. 

It  follows  from  this  series,  that  of  the  two  images  formed  by  double  refraction  in  Vesuvian,  and  other  similar 
crystals,  the  most  refracted  should  be  the  least  dispersed,  a  peculiarity  we  have  not  yet  had  an  opportunity  01 
Terifying  by  direct  observation.  It  follows,  however,  immediately  from  the  theory  of  the  rings  above  delivered, 
since  the  smaller  the  diameters  of  the  rings  for  any  coloured  ray,  the  greater  the  separation  of  its  pencils  by 
double  refraction.  Hence,  in  the  present  case,  the  red  rays  will  be  separated  by  a  greater  interval  than  the  violet 
in  the  two  spectra ;  and,  consequently,  the  least  refracted  spectrum  will  be  the  longest.  In  the  variety  of 
apophyllite  exhibiting  white  and  black  rings,  (leucocydite)  the  two  dispersions  should  be  almost  exactly  equal, 
and  the  only  difference  between  the  two  spectra  ought  to  consist  in  a  slight  variation  in  the  proportional  breadths 
of  the  several  coloured  spaces  in  them. 

Another  very  important  optical  character  is  the  intensity  of  the  polarizing,  or  doubly  refractive  energy.  This 
may  be  concluded  by  measuring  the  actual  angular  separation  of  the  images ;  but  this  is  usually  too  small  to 

VOL.  iv.  4  F 


1126. 


578 


LIGHT. 


physical 
character 
of  media. 


admit  of  being  determined  with  sufficient  precision,  in  such  very  imperfect  specimens  as  are  usually  subjected  to     Part  IV. 
examination  for  the  purpose  of  identification,  and  a  much  better  course  is  to  make  the  tint  developed  at  a  per-  ^— *v— 
pendicular  incidence,  by  a  plate  of  given  thickness  in  a  direction  at  right  angles  to   both   the  optic   axes,  the 
object  of  determination.     This  tint  (which  we  shall  term  the  equatorial  tint)  may  be  derived  immediately  from 
observations  of  tints  at  any  angle,  by  the  formula 


COS  f 


t     '  sin  0  .  sin  ff  ' 

where  N  is  the  tint  in  question,  numerically  expressed  as  usual,  and  where  n  is  the  tint,  (also  similarly  expressed) 
developed  at  an  angle  of  incidence  whose  corresponding  angle  of  refraction  is  p,  on  a  plate  whose  thickness  is  t, 
(expressed  in  English  inches  and  decimals)  and  where  0,  0'  are  the  angles  made  by  the  ray  in  traversing  the  plate 

with  the  two  axes.     This  value  of  N  is  the  same  with  —  in  the  equation  of  Art.  907.     The  following  list  of  a 

A 

very  few  substances  will  suffice  to  show  the  great  range  the  value  of  N  admits,  and  its  consequent  utility  as  a 
physical  character,  considerations  which  we  hope  will  induce  observers  to  extend  the  list  itself,  as  well  as  to 
give  it  all  possible  exactness. 


UNIAXAL  CRYSTALS. 


For  mean  yellow  rays. 


N  = 
35801 

—  r 

0.000028 

1246 

0.000802 

851 

0.001175 

470 

0.002129 

312 

0.003024 

109 

0.009150 

101 

0.009856 

41 

0.024170 

33 

0.030374 

Ditto.           3d  variety  .  . 

3 

0.3Cfi6'20 

BIAXAL  CRYSTALS. 


Nitre           .         

For  mean 

Ms 

7400 
1900 
1307 
521 
249 

yellow  rays. 

0.000135 
0.000526 
0.000765 
0.001920 
0.004021 

Anhydrite  (angle  between  axes  —  43°  48')                                

Heulandite  Cwhite:  —  anele  between  axes  1=54°  17').  .                 ... 

1127. 

Use  of  pola- 
rized light 
in  detect- 
ing complex 
structures. 


1128. 
Compound 
crystals  of 
nitre. 
Arragonite. 

1129. 

Topaz. 


1130. 

Tesselite. 
Fig.  223. 


But  the  phenomena  of  refraction,  reflexion,  and  polarization,  may  not  only  be  applied  by  the  aid  of  these  and 
similar  tables  of  registered  results,  to  the  examination  and  identification  of  substances  in  the  gross,  they  are  also  of 
use  in  detecting  peculiarities  of  structure  in  individual  specimens,  or  in  certain  species  which  would  otherwise 
escape  observation.  The  singular  structure  of  amethyst  has  been  already  explained,  and  a  variety  of  cases  of 
hemitropism  might  be  noticed,  in  which  the  juxtaposition  of  the  parts  is  rendered  evident  by  the  test  of  polarized 
light.  Of  these,  however,  by  far  the  most  curious  and  interesting  are  those  in  which  the  juxtaposed  parts  com- 
bine to  form  a  regular  whole,  and  to  produce  a  species  of  pseudo-crystal,  built  up  as  it  were  of  several  individuals, 
arranged  with  a  regard  to  symmetry,  and  forming  a  structure  of  more  or  less  complication.  Such  instances 
have  been  more  particularly  noticed  in  nitre,  arragonite,  topaz,  apophyllite,  sulphate  of  potash,  analcime,  har- 
motome,  &c. 

The  usual  form  of  the  crystals  of  nitre,  when  large  and  well  developed,  is  the  regular  hexagonal  prism ;  but  a 
section  of  this,  cut  at  right  angles  to  the  axis,  is  very  commonly  found  to  consist  of  two  or  more  portions,  in 
which  the  optic  meridians  are  60°  inclined  to  each  other  ;  but  the  plane  of  division  often  intersects  one  of  the 
lateral  faces  of  the  prism,  without  any  visible  external  mark  of  a  breach  of  continuity,  so  that  but  for  the  test  of 
polarized  light,  the  macled  structure  would  never  be  discerned.  The  phenomena  of  arragonite,  in  this  respect, 
are  very  similar  to  those  of  nitre. 

If  a  plate  of  Brazilian  topaz,  cut  at  right  angles  to  the  axis  of  the  rhombic  prism  in  which  it  crystallizes, 
be  examined  by  polarized  light,  it  will  occasionally  be  found  to  consist  of  a  central  rhomb,  surrounded  by  a 
border  in  which  the  optic  meridians  of  the  alternate  sides  are  inclined  at  £  of  a  right  angle  to  that  of  the  central 
compartment,  and  J  a  right  angle  to  each  other.  In  consequence,  when  such  a  rhombic  plate  is  held  with 
its  long  diagonal  in  the  plane  of  primitive  polarization,  two  opposite  sides  of  the  border  appear  bright,  the 
other  two  black,  and  the  central  compartment  of  intermediate  brightness.  Such  specimens  often  exhibit  the 
phenomena  of  dichroism  in  the  central  compartment,  while  the  border  is  colourless  in  all  positions. 

But  it  is  in  the  apophyllite  of  the  variety  named  by  Dr.  Brewster,  Tesselite,  that  this  enclosure  of  one  crystal 
in  a  case  as  it  were  of  another,  is  exhibited  in  the  most  regular  and  extraordinary  manner.  In  one  of  the  vane, 
ties  of  this  singular  body,  whose  form  is  the  right  rectangular  prism  with  flat  summits,  slices  taken  off  from  either 
summit  were  found  by  him  to  be  of  uniform  structure ;  but.  when  these  were  detached,  every  subsequent  slice  was 


L  I  G  H  T.  579 

found  to   consist  of  a  rectangular  border  enclosing  no  less  than   nine  several  compartments,  arranged  as  in     ['art  IV 
fig.  223,  and  separated  from  each  other,  and  from  the  border,  by  delicate  lines  or  films  as  there  marked.     Each  — -~*^~^> 
of  these  compartments  possesses  its  own  peculiar  crystallographic  structure,  and  polarizes  its  peculiar  tints,  the 
law  of  symmetry  being  observed.     In  some  specimens  the  triangular  spaces  p  q  r  s  were  wanting,  while  in  others 
they  seem  to  have  consisted  of  two  portions,  separated  by  an  imaginary  prolongation  of  the   line  joining  their 
obtuse  angles  with  the  central  lozenge. 

The  terminal  plates,  the  central  lozenge,  and  the  minute  stripes  dividing  the  compartments  from  each  other 
(which  are  sections  of  laminae  or  films  parallel  to  the  axis  of  the  crystal,  and  running  its  whole  length)  consist 
of  that  uniaxal  variety,  in  speaking  of  which  we  have  used  the  term  leucocyclite,  from  the  whiteness  of  its  rings. 
The  rectangles  R  V,  S  T,  (with  the  exception  of  the  portions  occupied  by  the  lozenge  and  partitions)  consist  of 
a  biaxal  medium,  having  its  axes  34°  inclined  to  each  other,  and  its  optic  meridian  parallel  to  the  axis  of  the 
prism,  and  passing  through  the  diagonals  R  V,,S  T  of  these  rectangles.  The  other  rectangles  are  composed 
of  a  similar  medium,  but  with  its  optic  meridian  at  right  angles  to  the  former,  or  passing  through  the  diagonals 
RT,  S  V. 

A  still  more  remarkable  and  artificial  structure  has  been  observed  by  Dr.  Brewster,  in  a  variety  of  the  Faroe      1131. 
apophyllites  of  a  greenish  white  hue.     When  a  complete  prism  of  this  variety  is  exposed  to  polarized  light,  with  Another 
its  axis  in  45°  of  azimuth,  the  light  being  transmitted  perpendicularly  through  two  opposite  sides,  the  pattern  varlety- 
represented  in  fig.  224  is  seen,  in  which  the  central  curvilinear  area  is  red,  and  its  complements  to  the  surround-  F'B-  *24. 
ing  rectangle  green.     The  squares  immediately  adjacent  on  either  side  in  the  direction  of  the  axis  are  also  vivid 
red  in  their  centres,  fading  into  white,  while  the  rest  of  the  pattern  consists  in  a  most  brilliant  succession  of  red, 
green,  and  yellow,  bands ,   for  a  coloured  figure  of  which  we  must  refer  the  reader  to  the  original  most  curious 
and  interesting  memoir,  (Edinburgh  Transactions,  vol.  ix.  part  ii.)  where,  as  also  in  the  Edinburgh  Philosophical 
Journal,  vol.  i.  he  will  find  the  phenomena  described  in  full  detail. 

The  sulphate  of  potash  offers  another  very  remarkable  example  of  compound  structure.     This  salt  occurs  in      1132. 
hexagonal  prisms,  and  occasionally  in  bipyramidal  dodecahedrons.     But  besides  these  forms  it  also  occurs  in  Sulphate  of 
rhombic  prisms  of  1 14°  and  66°.     These  Dr.  Brewster  found  to  have  two  axes,  while  the  hexagonal  prisms  have  Potash-. 
but  one;  thus  affording  another  instance  of  dimorphism  in  addition  to  those  of  arragonite,  sulphur,  &c.     On  ex- 
amining the  dodecahedrons,  however,  he  found  them  to  consist  of  six  equilateral  triangular  prisms,  of  the  biaxal 
variety,  grouped  together,  and  having  their  optic  meridians  all  converging  to  the  common  axis ;  the  molecules 
being  so  disposed  in   each  opposite  pair  of  individuals  as  to  make  the  angle  between  the  opposite  faces  of  either 
pyramid  (114°)  equal  to  the  obtuse  angle  of  the  rhomboid. 

The  structure  and  mode  of  action  of  the  analcime,  described  by  Dr.  Brewster  in  vol.x.  of  the  Edinburgh  Trans-  1 133. 
actions,  part  i.  p.  187,  are  so  extremely  singular,  that  it  is  difficult  to  say  whether  it  should  be  regarded  as  a  Analcime. 
grouped  crystal,  consisting  of  independent  portions  adhering  together,  or  as  amass  the  distribution  of  the  ether  in 
whose  parts  is  governed  by  a  general  and  uniform  law  ;  the  latter,  however,  is  probably  the  truth.  The  form  of 
this  crystal  is  the  icositetrahedron,  contained  by  twenty-four  similar  and  equal  trapezia,  and  may  be  regarded  as 
derived  from  the  cube  by  the  truncation  of  each  of  its  angles  by  three  planes  symmetrically  related  to  the  edges 
including  it.  If  we  conceive  from  the  centre  of  this  cube,  (in  its  natural  situation  with  respect  to  the  derived 
figure)  planes  to  pass  through  each  of  the  edges,  and  through  each  of  the  diagonals  of  the  six  faces,  they  will 
divide  the  cube  into  twenty-four  irregular  tetrahedra ;  and  of  these,  all  the  faces  which  pass  through  edges  of  the 
cube  will  also  pass  through  edges  of  the  derived  figure,  while  those  which  pass  through  diagonals  of  faces  of  the 
cube  will  also  pass  through  diagonals  of  the  faces  of  its  derivative,  bisecting  their  obtuse  angles.  Now  it  appears 
from  Dr.  Brewster's  observations,  that  all  the  molecules  situated  in  any  part  of  any  one  of  these  planes  are  devoid 
of  the  power  of  double  refraction  and  polarization  ;  and  that  in  proportion  as  a  molecule  is  distant  from  all  such 
planes,  its  polarizing  power  is  greater.  In  this  respect  it  differs  entirely  from  all  crystals  hitherto  examined, 
every  particle  of  which,  wherever  situated,  so  long  as  they  belong  to  one  and  the  same  crystalline  system,  being 
equally  endued  with  the  polarizing  virtue.  Nor  is  there  a  closer  analogy  between  the  mode  of  action  in  question, 
and  that  of  unannealed  glass  and  similar  bodies  ;  for  in  these  a  change  of  external  form  is  always  accompanied  with 
a  change  of  the  polarizing  powers,  while  in  the  analcime  each  particular  portion,  whether  detached  from  the  mass, 
or  in  its  natural  connection  with  the  adjacent  molecules,  possess  the  very  same  optical  properties.  The  action 
too  of  the  portions  which  possess  a  polarizing  power  is  not  related  to  axes  given  only  in  direction,  and  passing 
through  every  molecule,  but  to  planes  given  both  in  direction  and  in  place  within  the  mass,  (the  planes  above 
mentioned  ;)  the  tint  developed  at  any  point  of  a  plate  being  as  the  square  of  the  distance  from  the  nearest  of  such 
planes,  and  the  isochromatic  lines  being,  in  consequence,  straight  fringes  of  colour  arranged  parallel  to  the  dark 
bands  marked  out  by  the  intersection  of  such  planes  with  the  plate  examined.  The  phenomena  described  are 
accompanied  with  a  sensible  double  refraction.  The  reader  is  referred  to  the  memoir  already  cited  (which  is  one 
of  the  most  interesting  to  which  we  can  direct  his  attention)  for  further  details :  and  to  a  work  understood  to  be 
forthcoming  from  the  pen  of  the  eminent  author  here  and  so  often  before  cited,  on  optical  mineralogy,  for  what 
we  are  sure  will  prove  a  treasure  of  valuable  information  on  every  point  connected  with  this  important  application 
of  optical  science 

§  XIV.  On  the  Colours  of  Natural  Bodies. 

It  was  onr  intention  to  have  devoted  a  considerable  share   of  these  pages  to  the  explanation  of  such  natural      H34. 
phenomena  as   depend  on  optical  principles,  but  the  great  length  to  which  this  essay  has  already  extended,  renders 
it  necessary  to  confine  what  we  have  to  say  on  such  subjects  within  very  narrow  limits,  and  to  points  of  promi- 

4  F  2 


580  L  I  G  H  T. 

Light.      nent  importance.     Among  these  there  is  certainly  none  more  entitled  to  consideration  than  the  phenomena  of     Pan  IV. 
v>— -v"»«'  colour,  as  exhibited  by  natural  objects,  which  strike  us  wherever  we  turn  our  eyes,  and  it  is  impossible  to  pass  in  s— v^— 
Newton's      total  silence  the  theory  devised  by  Newton  to  account  for  them ;  a  theory  of  extraordinary  boldness  and  subtilty, 
^tlle  in  which  great  difficulties  are  eluded  by  elegant  refinements,  and  the  appeal  to  our  ignorance  on  some  points  is 
natural         so  dexterously  backed  by  the  weight  of  our  knowledge  on  others,  as  to  silence,  if  not  refute,  objections  which  at 
bodies.          first  sight  appear  conclusive  against  it.     The  postulates  on  which  this  theory  rests  are  essentially  as  follows : 

1135  1.  All  bodies  are  porous  ;  the  pores  or  intervals  vacant  of  ponderable  matter,  occupying  a  very  much  larger 

Postulates,    portion  of  the  whole  space  filled  by  the  body,  than  the  solid  particles  of  which  it  essentially  consists. 

1136.  2.  These  so'.id  particles  have  a  certain  size  (and  perhaps  figure)  essential  to  them  as  particles  of  that  particular 
medium,  and  which  cannot  be  changed  by  any  mechanical  action,  or  by  any  means  not  involving  a  change  in  the 
chemical  nature  or  condition  of  the  medium.  They  are,  in  short,  the  ultimate  atoms;  to  break  which,  is  to  destroy 
their  essence,  and  resolve  them  into  other  forms  of  matter,  having  other  properties. 

1137.  3.  These  atoms  are  perfectly  transparent,  and  equally  permeable  to  light  of  all  refrangibilities,  which,  having 
once  passed  their  surfaces,  is  in  the  act  of  pursuing  its  course  through  their  substances. 

Newton,  indeed,  makes  his  atoms  only  "  in  some  measure  transparent."  But  he  never  refers  to  this  limitation, 
and  his  theory  depends  essentially  on  their  perfect  transparency,  as  is  indeed  obvious  from  his  account  of  opacity, 
which  is  contained  in  the  next  postulate. 

1 138.  4.   Opacity  in  natural  bodies  arises  from  the  multitude  of  reflexions  cawed  in  their  internal  parts. 

Cause  of  jt  ;s  obvious,  therefore,  that  unless  we  admit  a  cause  of  opacity  in  atoms  different  from  that  which,  on  this 

opacity.  hypothesis,  causes  it  in  their  aggregates  constituting  natural  bodies,  the-former  cannot  be  otherwise  than  abso- 
lutely pellucid,  since  no  reflexions  can  take  place  where  there  are  no  intervals,  and  no  change  of  medium.  Of 
the  sufficiency  of  this  cause,  either  in  natural  bodies  or  atoms,  however,  we  confess  there  does  appear  to  us  some 
room  for  doubt,  as  it  seems  difficult  so  to  conceive  these  internal  reflexions,  that  the  rays  subjected  to  them  shall 
be  all  andybr  ever  retained,  entangled  as  it  were,  and  running  their  rounds  from  atom  to  atom,  without  a  possi- 
bility of  reaching  the  surface  and  escaping  ;  which,  were  they  to  do,  it  is  evident  that  every  body  so  con- 
stituted, receiving  a  beam  of  light,  would  in  fact  only  disperse  it  in  all  directions  in  the  manner  of  a  self 
luminous  one. 

1 139.  5.  The  colours  of  natural  bodies  are  the  colours  of  thin  plates,  produced  by  the  same  cause  which  produces  them 
Origin  of      in  thin  lamince  of  air,  glass,  fyc.  viz.  the  interval  between  the  anterior  and  posterior  surfaces  of  the  atoms,  which, 
natural         when  an  odd  multiple  of  half  the  length  of  a  fit  of  easy  reflexion  and  transmission  for  any  coloured  ray  moving 

r3'  within  the  medium,  obsti  u  cts  its  penetration  of  the  second  surface,  and  when  an  even,  ensures  it,  (see  Art.  655.)  The 
thickness,  therefore,  of  the  atoms  of  a  medium,  and  of  the  interstices  between  them,  determines  the  colour  they 
hhall  reflect  and  transmit  at  a  perpendicular  incidence.  Thus,  if  the  molecules  and  interstices  be  less  in  size 
than  the  interval  at  which  total  transmission  takes  places,  or  less  than  that  which  corresponds  to  the  edge  of  the 
central  black  spot  in  the  reflected  rings,  a  medium  made  up  of  such  atoms  and  interstices  will  be  perfectly  trans- 
parent. If  greater,  it  will  reflect  the  colour  corresponding  to  its  thickness. 

1 140.  It  may  be  objected  to  this,  that  all  natural  colours  do  not  of  necessity  find  a  place  in  the  scale  of  tints  of  thin 
Objections,  plates,  even  those  of  bodies  whose  chemical  composition  is  uniform  ;  but  to  this  we  may  answer,  that  the  colours 

reflected  from  the  first  layer  only  of  molecules  .next  the  surface  ought  to  be  pure  tints,  those  fiom  lower  layers 
having  to  make  their  way  to  the  eye  through  the  upper  strata,  and  thus  undergoing  other  analyses,  by  trans- 
missions and  reflexions  among  the  incumbent  atoms.  Besides  which,  whatever  shape  we  attribute  to  the  a'.oms, 
it  is  impossible  that  all  rays  shall  penetrate  them  so  as  to  traverse  the  same  thickness  of  thorn,  unless  we  regard 
them  as  mere  lamince  without  angles  or  edges,  and  of  enormous  refractive  power.*  The  same  answer  must  be 
made  to  the  objection,  equally  obvious,  that  the  transmitted  tint  ought  to  be  in  all  coses  complementary  to  the 
reflected  one,  and  that  therefore  cases  like  that  of  leaf  gold,  opalescent  glass,  and  infusion  of  lignum  nqihriti- 
cum,  all  which  reflect  one  tint  and  transmit  another,  but  in  all  which  this  condition  is  violated,  form  exceptions 
to  the  theory.  But,  in  reality,  the  transmitted  rays  have  traversed  the  whole  thickness  of  the  medium,  and  have 
therefore  undergone,  many  more  times,  the  action  of  its  atoms,  than  those  reflected,  especially  those  near  the 
first  surface,  to  which  the  brighter  part  of  the  reflected  colour  is  due. 

1141.  The  infusion  of  Ii<rnum  nephriticum  is  a  very  singular  case,  and  its  peculiar  properties  have  been  explained  by 
Apparent      Dr.  Young,  on  the  supposition  of  minute  particles  of  definite  magnitude  suspended  in  it.     Though  very  trans- 
exceptions    parent)  it  yet  reflects  a  bluish  green  colour,  while  the  light  transmitted  is  yellow  or  wine-coloured,  in  this  re- 

spect  offering  almost  the  exact  converse  of  leaf  gold.  It  is,  however,  no  doubt  a  case  of  opalescence,  and  is 
exactly  imitated  by  certain  yellow  glasses,  in  which  a  very  visible  thin  film  of  opalescent  matter  near  the  surface 
reflects  to  the  eye  a  bluish  green  tint,  while  yet  the  colour  transmitted  has  the  yellow  tint  belonging  to  the  glass. 
The  reflexion  proceeds  from  particles  which  have  nothing  to  do  with  the  transmitted  light. 

1142.  B"t.  in  foct.  *«e  objection  (as  appears  to  us)  is  not  yet  fully  answered.     Transparent  coloured  media  (clear 
Case  of        liquids  in  which  no  floating  particles  exist,)  have  no  reflected  colour.     When  examined  by  pouring  them  into  an 
transparent    opaque  vessel,  blackened  internally    and  filled  to   the   brim,   and  when   the    colourless  reflexion  from   their 
:uloured       upper  surface  ;s  destroyed  by  reflexion  in  an  opposite  plane   at  the  polarizing  angle,  it  is  seen  at  once  that 

no  light  is  reflected  from  within  the  medium,  either  near  the  surface,  or  at  greater  depths ;  and  if  this  mode  of 
examination  be  regarded  as  objectionable,  as  perhaps  destroying  the  internal  as  well  as  external  reflexion,  it  is 
equally  satisfactory  to  observe,  that  the  image  of  a  white  object  reflected  from  the  surface  of  a  fluid  in  a  black 
opaque  vessel  is  always  purely  white,  whatever  be  the  colour  of  the  reflecting  fluid.  We  are  not  aware  that  the 
objection  so  put  has  been  sufficiently  considered,  or  even  propounded.  To  us  its  weight  appears  considerable, 

*  Newton  appears  to  nave  been  fully  aware  of  the  necessity  of  taking  this  into  consideration.     Prop.  vii.  book  ii.  Opt.  versus  Jtnem. 


L  1  G  H  T. 

Light.      and  we  cannot  but  believe  that,  some  other  cause  besides   mere  internal  reflexions  must  interfere  to  prevent  the    Part  iv 
•N^^^  complementary  colour  from  reaching  the  eye;    and  that  absorption,  with  its  kindred  phenomenon,  or  rather  its  *— v1— ' 
extreme  case,  opacity,  is  not  satisfactorily  accounted  for  in  this  theory,  but  must  rather  be  admitted  as  (at  pre- 
sent,) an  ultimate  fact,  of  which  the  cause  is  yet  to  seek. 

If  this  be  granted,  the  colours  of  all  bodies  may  be  distinguished  into  true,  viz.,  those  which  arise  from  rays       1143. 
whirJi  have  actually  entered  their  s-ubstance  and  undergone  their  absorptive  action,  (as  the  colours  of  powders  True  ami 
of  transparent  coloured  media,  cinnabar,  red  lead,  Prussian  blue,  those  of  flowers,  &c.,)  and  false,  or  superficial,  false  Ca~ 
or  those  which  originate  obviously  in  the  law  of  interference  ;  thus,  the  variable  colours  of  feathers,  insects'  wings,    uurs' 
striated  surfaces,  oxidated  steel,  and  a  variety  of  cases  to  which  the  Newtonian  doctrine  strictly  applies,  for  there 
is  no  denying:  that  cases  of  colour,  not  merely  superficial,  do  occur,  in  which  the  Newtonian  doctrine,  to  say  the 
least,  is  highly  probable.     To  instance  one  or   two  only.     If  a  few  drops  of  an  extremely  weak  solution  of  Cases  in 
nitrate  of  silver  be  added  to  a  very  dilute  solution  of  hyposulphite  of  lime,  a  precipitate  is  formed  of  an  opales-  whl,ch*^ 
cent  whiteness  and  extreme  tenuity.     If  more  of  the  nitrate  be  added,  the  precipitate  increases  in  weight  and  appj;eg 
aggregation,  and  at  the  same  time  changes  its  colour,  becoming  first  yellow,  then  yellow  brown,  then  a  rich 
orange  brown,  then  a  purplish  brown,  and,  finally,  a  deep  brown  black.     The  precipitate,  meanwhile,  continually 
acquires  density,  and,  finally,  sinks  rapidly  to  the  bottom.     It  is  impossible,  in  this  series,  not  to  trace  the  tints 
of  the  first  order  of  reflected  rings,  produced  by  the  thickening  of  the  minute  particles  in  the  act  of  aggregation, 
but  equally  impossible  not  to  recognise  the  agency  of  a  cause  totally  different,  acting  to  increase  the  opacity  of 
the  compound  by  an  absorptive  action  far  superior  to,  and  independent  of,  the  action  of  the  particles  as  thin 
plates.     The  phenomena  of  Hematine,  described  by  Chevreul  and  cited  by  Dr.  Brewster,  (Encyc.  Edin.  Optics, 
p.  623  ;    see  also  Biot,  Traite  de  Pfiys.  torn  iv.  p.  134,  there  referred  to,)  afford  too  close  an  approximation  to 
the  series  of  tints  of  the  second  order  not  to  authorize  a  presumption  that  the  Newtonian  theory  may  apply  to 
this  case  also.     The  diffused  light  and  blue  colour  of  the  clear  sky,  affords  another  very  satisfactory  instance. 
This  blue  is,  no  doubt,  a  blue  of  the  first  order,  reflected  from  minute  aqueous  particles  in  the  air.     The  proof 
is,  that  at  74°  distance  from  the  sun,  it  is  completely  polarized  in  a  plane  passing  through  the  sun's  centre. 

Another  objection,  no  less  obvious,  to  the  Newtonian  doctrine,  has  been  successfully  answered  by  Newton      1144 
himself.     A  change  of  obliquity  of  incidence,  it  may  be  urged,  should  cause  a  change  of  colour,  as  a  plate  of  Another  ulj- 
given  thickness  reflects  a  different  tint  at  oblique  and  perpendicular  incidences.     But  this  variation  is  less,  the  jection. 
greater  the  refractive  power  of  the  medium  ;    and  as  the  refractive  power  increases  with  the  density,  that  of  the  ADSwered- 
"Jense  ultimate  atoms  of  bodies  must  be  exceeding  great,  so  that  the  tint  reflected  from  them  will  vary  little  with 
i  change  of  incidence,  (art.  669.)     The  colours  of  oxidated  steel  afford  an  excellent  case  in  point.    The  refractive 
power  of  this  oxide,  though  great,  (2.1),  is,  doubtless,  not  to  be  compared  with  that  of  the  ultimate  atoms  of 
bodies,  yet  the  tints  on  the  surface  of  blued  steel  vary  but  little  with  a  change  of  obliquity.     We  may  add,  too, 
that  the  colour  exhibited  by  any  body  of  sensible  magnitude,  is  in  reality  an  average  of  the  colours  reflected 
from  all  its  molecules  at  all  possible  incidences,  so  that  no  change  of  incidence  ought  to  be  expected  to  atlect  it. 

Of  the  extreme  tenuity  of  the  ultimate  molecules  of  bodies,  Newton  seems  to  have  had  but  an  inadequate      1145 
\dea,  as  he  supposed  that  they  might  be  seen  through  microscopes  magnifying  three  or  four  thousand  times.*  Newton's 
We  have  viewed  an  object  without  utter  indistinctness,  through  a  microscope  by  Amici,  magnifying  upwards  of  iJeas  of  the 
three  thousand  times  in  linear  measure,  and  had  no  suspicion  that  the  object  seen  was  even  approaching  to  s'ze.°*""!- 
resolution  into  its  primitive  molecules.     But  it  should  rather  seem  that  Newton  regarded  his  colorific  molecules  {Jj 
as  divisible  groupes  of  atoms  of  a  yet  more  delicate  kind,  and  yet  more  densa,  and  these  again  as  still  further 
resolvable  till  the  last  stage  of  indivisibility  be  reached.     M.  Biot  has  given  a  striking,  and,  we  may  almost  term 
it,  picturesque  account  of  this  doctrine,  in  his  Traite  de.  Physique. 

§  XV.    Of  the  Calorific  and  Chemical  Rays  of  the  Solar  Spectrum. 

It  has  long  been  a  matter  of  everyday  observation,  that  solar  light  exercises  a  peculiar  influence  in  altering 
the  colours  of  bodies  exposed  to  it,  either  by  deepening  or  discharging  them,  even  when  totally  secluded  from 
air,  and  that  various  metallic  salts  and  oxides,  especially  those  of  silver,  are  speedily  blackened  and  reduced 
when  freely  exposed  to  direct  sunshine,  or  even  to  the  ordinary  light  of  a  bright  day.  Whether  these  effects 
were  owing  to  the  heat  of  the  rays,  or  to  some  other  cause,  remained  long  uninquired.  The  first  step  was 

*  The  passage,  however,  is  in  the  highest  tone  of  a  refined  philosophy,  and,  mdepeodenl  of  its  theoretic  bearings,  we  extract  it,  as 
indicating  a  scrutinizing  spirit  of  observation  far  beyond  the  age  he  lived  in. 

"  In  these  descriptions  I  have  been  the  more  particular,  because  it  is  not  impossible  but  that  m'croscopes  may  at  length  be  improved  to  the 
discovery  of  the  particles  of  bodies  on  which  their  colours  depend,  if  they  are  not  already  in  some  measure  arrived  to  that  degree  of  perfection. 
For  if  those  instruments  are  or  can  be  so  far  improved  as  with  sufficient  distinctness  to  represent  objects  five  or  six  hundred  times  bi^rer 
than  at  a  foot  distance  they  appear  to  our  naked  eyes,  I  should  hope  that  we  might  be  able  to  discover  some  of  the  greatest  of  those  cor- 
puscles. And  by  one  that  would  magnify  three  or  four  thousand  times  perhaps  the)  might  all  be  discovered,  but  those  which  produce 
blackness.  In  the  mean  while  I  see  nothing  material  in  this  discourse  that  may  rationally  be  doubted  of,  excepting  this  position  :  That 
transparent  corpuscles  of  the  same  thickness  and  density  with  a  plale,  do  exhibit  the  same  colour.  And  this  I  would  have  understood  not 
without  some  latitude,  as  well  because  those  corpuscles  may  be  of  irregular  figures,  and  many  rays  must  be  obliquely  incident  on  them,  and 
so  have  a  shorter  way  through  them  than  the  length  of  their  diameters,  as  because  thestraitness  of  the  medium  put  in  on  all  sides  within 
such  corpuscles  may  a  little  alter  its  motions  or  other  qualities  on  which  the  reflection  depends.  But  yet  I  cannot  much  suspect  the  last 
because  I  have  observed  of  some  small  plates  of  Muscovy  glass  which  were  of  an  even  thickness,  that  through  a  microscope  they  have 

id  corners  where  the  included  medium  was  terminated,  whic1   " 

;,  if  those  corpuscles  can  be  discovered  with  microscopes  ;  w 

is  sense.     For  it  seems  imoossible  to  see  the  more  secret  am 
.orpuscles,  by  reason  of  their  transparency." 


58-2 


LIGHT. 


Light. 
s— v— 
Diir.ovenes 
of  Scheele, 
Sir  W. 
Heischel, 

Ritter. 


1147. 

Calorific, 
luminous, 
and  chemi- 
cal rays. 


1148. 

All  obey  the 
same  optical 
laws. 

Chemical 
rays  inter- 
fere like  lu- 
minous 
ones. 

1149. 

Wjllaston's_ 
observa- 
tions on 

iuiacum. 


1150. 

Effect  of 
-ight  on 
purple 
glass. 

1151. 
Olher  ef- 
fects of  50- 
hr  light. 


1152. 


made  by  Scheele,  who  ascertained  that  muriate  of  silver  is  much  more  powerfully  blackened  in  the  violet  rays 
than  in  any  other  part  of  the  spectrum.  (Traits  de  I' Air  et  du  Feu,  §  66.)  The  experiments  of  Sir  W.  Herschel, 
on  the  heating  power  of  the  several  prismatic  rays,  on  the  other  hand,  which  appeared  in  1800,  showed  satis- 
factorily that  the  more  refrangible  rays  possess  very  little  heating  power,  the  calorific  effect  being  at  its  maxi- 
mum for  the  .extreme  red  rays,  and  even  extending  considerably  beyond  the  limits  of  the  spectrum  in  that 
direction.  This  remarkable  discovery,  which  established  the  independence  of  the  heating  and  illuminating 
effects  of  the  solar  rays,  led  Professor  Ritter,  of  Jena,  in  1801,  to  examine  whether  a  similar  extension  beyond 
the  limits  of  the  visible  spectrum  might  not  also  have  place  in  the  chemical  or  deoxidating  rays,  and  on  exposing 
muriate  of  silver  in  various  points  within  and  without  the  spectrum,  he  found  the  maximum  of  effect  to  lie 
beyond  the  visible  violet  rays,  the  action  being  less  in  the  violet  itself,  still  less  in  the  blue,  and  diminishing 
with  great  rapidity  as  he  proceeded  towards  the  less  refrangible  end.  Dr.  Wollaston  independently  arrived  at 
the  same  conclusion. 

The  solar  rays,  then,  possess  at  least  three  distinct  powers  :  those  of  heating,  illuminating,  and  effecting 
chemical  combinations  or  decompositions,  and  these  powers  are  dislributed  among  the  differently  refrangible 
rays,  in  such  a  manner  as  to  show  their  complete  independence  on  each  other.  Later  experiments  have  gone  a 
certain  way  to  add  another  power  to  the  list — that  of  exciting  magnetism.  \\  ithout  calling  in  question  the 
accuracy  of  the  observations  which  are  directed  to  establish  this  point,  we  may  be  permitted  to  hope  that  further 
researches  will,  ere  long,  explain  the  causes  of  failure  in  those  numerous  cases  where  such  effects  have  not  been 
produced. 

The  calorific  rays  appear,  from  experiments  of  Berard,  to  obey  the  laws  of  polarization  and  double  refraction, 
like  those  of  light.  Those  of  interference  could  not  be  made  without  excessive  difficulty.  In  the  case  of  the 
chemical  rays,  the  same  difficulty  is  not  experienced  ;  and  Dr.  Young,  and  after  him,  by  more  delicate  means, 
M.  Arago,  have  satisfactorily  demonstrated  that  these  conform  to  the  same  laws  of  interference,  whether  po- 
larized or  otherwise,  that  are  obeyed  by  the  luminous  rays  similarly  circumstanced.  Thus,  a  set  of  fringes 
formed  by  the  interference  of  two  solar  pencils  with  a  common  origin,  being  kept  very  steadily  projected  for  a 
long  time  on  one  and  the  same  part  of  a  sheet  of  paper  rubbed  with  muriate  of  silver,  a  series  of  black  lines 
became  traced  on  it,  the  intervals  of  which  were  smaller  than  those  of  the  dark  and  luminous  fringes  formed  by 
homogeneous  violet  light. 

Dr.  Wollaston  having  observed  that  gum  guiacum  is  turned  green  by  exposure  to  solar  light  in  contact  with 
air,  took  two  specimens  of  paper  coloured  wilh  a  yellow  solution  of  this  gum  in  alcohol,  and  exposed  one  of 
them  to  air  and  sunshine,  the  other  to  air  in  the  dark.  The  former  was  turned  perceptibly  green  in  five  minutes, 
and  the  change  was  complete  in  a  few  hours,  while  the  latter  was  no  way  discoloured  after  many  months.  He 
then  concentrated  the  violet  rays  on  paper  so  coloured,  by  a  lens,  and  the  change  was  speedily  performed,  while 
in  the  most  luminous  there  was  no  change  of  colour.,  and,  in  the  red  rays,  the  green  colour  was  not  only  not 
produced,  but  when  induced  by  exposure  to  the  violet,  was  again  destroyed,  and  the  original  yellow  colour 
restored.  This  seems,  however,  to  have  been  merely  an  effect  of  the  heat,  as  the  warmth  from  the  back  of  a 
heated  silver  spoon  discharged  the  green  colour  just  as  effectually. 

Mr.  Faraday  has  observed  that  glass  tinged  purple  with  manganese,  has  its  hue  much  deepened  by  the 
passage  of  solar  light  through  it,  and  that  two  portions  of  the  same  plate,  one  preserved  in  the  dark,  the  other 
exposed  freely,  after  some  time  differ  materially  in  intensity  of  colour. 

The  direct  action  of  solar  light,  or,  possibly,  of  its  heat  also,  produces  other  chemical  effects,  such  as  the 
immediate  combination  of  the  elements  of  phosgen,  the  explosion  of  an  atomic  mixture  of  chlorine  and  hydrogen, 
and  other  phenomena,  all  indicative  of  powers  resident  in  this  wonderful  agent,  of  which  we  have  but  a  very 
imperfect  notion  at  present.  The  green  colour  of  plants,  and  the  brilliant  hues  of  flowers,  depend  entirely  on 
it.  Tansies  which  had  grown  in  a  coal  pit,  were  found  totally  destitute  either  of  colour  or  of  their  peculiar  and 
powerful  flavour,  and  the  bleaching  and  sweetening  of  celery  by  the  exclusion  of  light,  is  another  familiar  in- 
stance of  the  same  cause.  How  far  the  differently  coloured  rays  are  concerned  in  these  effects,  has  never  yet 
been  accurately  investigated,  though  attempts  have  been  made ;  but  we  hope,  from  the  distinguished  ability  of 
an  eminent  individual  who  has  recently  taken  up  this  most  interesting  inquiry,  that  our  stock  of  knowledge  will 
soon  receive  material  accessions. 

We  cannot  close  this  Essay  without  an  expression  of  regret,  that  the  Memoir  of  Professor  Airey,  on  the 
Spherical  Aberration  of  the  Eyepieces  of  Telescopes,  just  on  the  point  of  publication  in  the  Transactions  of  the 
Cambridge  Philosophical  Society,  reached  us  too  late  to  allow  of  our  attempting  to  condense  its  valuable  con- 
tents, and  we  can  only  recommend  it  to  the  notice  of  our  readers  in  lieu  of,  and  in  preference  to,  anything  we 
could  ourselves  say  on  that  subject.  A  similar  expression  of  regret  applies  to  the  interesting  "  Theory  of  Sys- 
tems of  Rays,'1  by  Professor  Hamilton  of  Dublin,  a  powerful  and  elegant  piece  of  analysis,  communicated  to  the 
Royal  Irish  Academy  in  1824,  and  only  now  in  the  course  of  impression,  but  of  which  enough  has  reached,  us, 
by  the  kindness  of  its  Author,  to  make  us  fully  sensible  of  the  benefit  we  might  have  derived  from  its  perusal  at 
an  earlier  period  of  our  undertaking. 


Part  IV. 


Slough,  December  12;  1827 


J.  F.  W.  HERSCHEL 


LIGHT. 


583 


INDEX. 


N.  B.   The  Numbers  are  those  of  the  Articles  as  they  stand  on  the  Margin. 


Light.  Aberration,  of  Light,  10.  Spherical,  for  reflected  rays,  128. 
j-  -n_-  Circle  of  least,  154. 156.  Of  a  system  of  surfaces  for  refracted 
rays,  281.  291.  Of  a  thin  single  lens,  293.  Its  comparative 
amount  in  different  lenses,  807.  Of  lenses  generally,  297.  Of 
a  system  of  thin  lenses,  308.  Its  effect  in  lengthening  or 
shortening  focus,  289.  General  equations  for  its  destruction, 
3!2,  S13. 

Aberration,  Chromatic,  explained,  456.  Circle  of  least,  457. 
Principles  of  its  destruction,  459. 

Absorption  of  Light  by  nncrystallized  media,  481,  et  seq.;  by 
crystallized,  1059,  ft  seq. 

Acht  omaticity,  general  conditions  of,  459. 

Achromatic  refraction,  427.  448.  Its  general  conditions,  459.  At 
common  surface  of  two  media,  478.  Produced  by  combina- 
tions of  one  medium,  451. 

Achromatic  Telescope,  theory  of,  456,  et  seq. 

Adaptation  of  the  eye  to  different  foci,  356. 

Amethyst,  its  peculiar  structure,  10U. 

A MICI,  his  prismatic  telescope,  453.     His  microscopes,  1115. 

Amplitude  of  an  undulation,  605. 

Analcime,  peculiar  polarization  produced  by,  1133. 

Analysis  of  solar  light  by  the  prism,  397.  406.  By  coloured 
glasses,  506.  Of  the  colours  of  thin  plates,  644. 

Angle  of  polarization,  831. 

Apertures,  waves  transmitted  through, '  631.  Phenomena  of 
diffraction  through,  729.  Of  telescopes,  of  different  forms, 
their  effect,  768. 

Apophyllite,  peculiar  rings  exhibited  by  its  several  varieties,  915. 
918.  Biaxal,  1130.  Variety  called  Tesselite,  its  structure, 
1130,  1131. 

AHA  co,  M  ,  his  mode  of  measuring  refractive  indices,  733.  His 
law  of  polarization  by  oblique  transmission,  947.  His  disco- 
very of  the  rotatory  phenomena  in  quartz,  1037.  His  laws  of 
interference  of  polarized  rays,  917. 

Axel  defined,  783.  Optic,  889.  Differ  for  differently  coloured 
rays,  921.  Their  situations  calculated  a  priori,  1008. 

Axes  of  elasticity,  1000.  Polarizing,  Brewster's  theory  of  their 
composition  and  resolution,  1020.  Of  double  refraction,  781. 
Positive  and  negative,  1021.  1032. 

BIOT,  M.,  his  doctrine  of  movable  polarization,  928.  His 
apparatus  described,  929.  His  researches  on  the  rotatory 
phenomena,  1037.  1045.  His  law  of  the  isochromatic  lines 
in  biaxal  crystals,  907.  His  rule  for  determining  the  planes  of 
polarization  within  biaxal  crystals,  1070. 

BLAIR,  Dr.,  his  achromatic  telescopes  with  fluid  object  glasses, 
474. 

/ilin<lnesst  its  causes  and  remedies,  360. 

How,  coloured  prismatic,  555,  556. 

BREWSTER,  Dr.,  his  law  of  polarization  by  reflection,  831. 
Laws  of  polarization  by  oblique  transmission,  866.  His 
optical  researches  and  discoveries,  passim.  His  theory  of 
polarizing  axes,  1020. 

Brightness,  intrinsic  and  absolute,  29.  See  Photometry.  Of 
Images,  349. 

Calorific  rays  of  the  solar  spectrum,  1 147. 

Camera  obscura,  330. 

Cassia,  oil  of,  its  remarkable  refractive  and  dispersive  powers, 

1117.1121.     Experiment  upon,  1122. 
Catacaustics,  or  Caustics  by  reflexion,  1 34,  el  seq.     Their  length, 

144.     Determination   of    from    a   given  reflecting  curve,  137. 

Conjugate,  146.    Density  of  rays  in,  160. 
Caustics  by  refraction,  226,  el  seq.     Of  a  plane,  238. 
CHAULNES,  Due  de,  phenomena  observed  by  him,  687. 
Chemical  rays  of  the  spectrum,  1 1 46,  el  seq. 
Chromatics,  395.     Chromatic  aberration.     See  Aberration. 
Circular  polarization,  10S7.  eiseq.     Vibrations,  627. 
CLAIRAUT,  his  condition  for  construction  of  achromatic  object 

glasses,  467. 


Coloured  rays  unequally  absorbed  by  media,  486. 

Coloured  rings  and  fringes.     See  Rings  and  Fringes. 

Colours  of  natural  bodies  not  inherent,  410.  Newton's  theory 
of  such  colours,  1 131,  et  seq.  Of  the  prismatic  spectrum,  421. 
Of  flames,  521.  Of  thin  plates,  633.  Of  thick  plates,  676. 
Of  mixed  plates,  696.  Of  fibres  and  striated  surfaces,  7UO. 

Colours,  primary,  Mayer's  hypothesis  respecting,  509.  Young's, 
518. 

Colours  polarized  by  crystallized  plates,  881. 

Colours,  periodical,  6S5",  «(  spr/.     True  and  false,  1 143. 

Composition  anil  resolution  of  vibrations,  620.     Of  axes,  1020. 

Cord,  stretched,  analogy  between  its  vibrations  and  those  of  ttie 
ether,  977.  980.  98R. 

Cornea  of  the  eye,  350.  Case  of  malconformation  of,  remedied, 
S58,  359. 

Crack  in  a  heated  glass  plate,  its  effect  on  the  polarized  tints, 
1102. 

Crested  fringes  observed  by  Grimaldi,  728. 

Cross,  black,  traversing  the  polarized  rings.  Its  form  in  uniaxal 
crystals,  911.  In  biaxal,  1092. 

Crystals,  Oniaxal,  enumerated,  785.  1121.  Law  of  double 
refraction  in,  795.  Uiaral,  table  of  the  inclinations  of  their 
axes,  1121.  Phenomena  of  the  polarized  lemniscates  ex- 
hibited by,  892,  et  teq.  1069,  et  teq.  General  law  of  double 
refraction  in,  101 I,  el  seq.  Action  of  heat  on,  1 109.  Positive 
and  neaatire,  or  attractive  and  repulsive,  803.  942.  How  dis- 
tinguished, 945. 

Crystallized  surfaces,  their  action  on  reflected  light,  1 123. 

Crystalline  of  the  eye,  852. 

Deflexion  of  light.     Newton's  doctrine  of,  713. 

Depolarization  of  light,  925. 

Depolarizing  axes,  1087. 

Deviation  of  a  ray  a  'ter  any  refraction  in  one  plane,  211.  Mini- 
mum produced  by  a  prism,  216.  Of  tints  from  Newton's  scale 
in  the  polarized  rings,  915.  1125. 

Diacauslics.     See  Caustics  by  refraction. 

Dichroism,  phenomena  of,  in  uniaxal  crystals,  10fi4.  In  biaxal, 
1067.  Expressed  by  an  empirical  formula,  1073. 

Dichromatic  media,  499. 

Diffraction  of  light,  706,  etsea. 

Dilatation  of  rings  at  oblique  incidences,  639.  6fi9.  Of  the 
diffracted  fringes  by  approach  of  the  radiant  point,  711.  Of 
glass,  its  effect  in  imparting  the  polarizing  property,  1089.  Of 
jellies,  1094. 

Discs,  spurious,  of  stars,  767. 

Dispersion  of  light,  395,  &c. 

Dispersire  powers  of  media,  425  Methods  of  determining  them, 
428.431.435.  A  very  precise  practical  one  for  object  glasses. 
483.  Table  of,  1120.  Of  higher  orders,  446. 

Due  de  Chaulnes,  his  experiment  on  coloured  rings,  687. 

Klnntic  forces  of  a  medium  generally  expressed,  998. 

Elasticity,  a\es  of,  1000.     Surface  of,  1004. 

Klliptic,  vibrations  of  ethereal  molecules,  621. 

Emanation,  oblique,  law  of,  43. 

Ether,   its   vibrations  the  (hypothetical)    cause   of  light,  56S. 

Frequency  of  its  pulsations,  575.     See  Undulations. 
Extinction  of  light,  484.  1 1 38. 
Eye,  its  structure,  350.    Change  of  focus,  S56.     Of  fishes,  368. 

See  Vision. 

Field  of  view,  381. 

nimf,  interrupting,  in  crystals,  phenomena  exhibited  by,  10?8, 

el  *ffj. 

Fifsof  easy  reflexion  and  transmission,  526.  P51 
fired  lines   in  the  spectrum  described,  418.     Their   utility   in 

optical  determinations.  420. 
Flames,  coloured,  their  phenomena,  5  JO. 
Foci,  general  determination  for  any  curve  by  reflected  rays,  109. 

112.     In  a  sphere,  183.  250.    Conjugate,  126.     General  inves- 


Incler. 


58-1 


LIGHT. 


Light.  tigation  of,  for  refracted  rays  in  any  curved  surface,  221.     In 

_ -v~uS  spherical  surface,  239,  et  seqr.  For  central  rays,  (fundamental 
equation,)  247.  Of  a  system  of  spherical  surfaces,  25S.  257. 
Of  a 'system  of  lenses,  268.  Of  thick  lenses,  272.  Of  doubly 
refractive  lenses,  805.  For  oblique  rays,  818,  etseq.  to  321. 
Aplanatic,  287.  How  conceived  in  the  undulatory  system,  590. 

FRAUNHUFER,  his  experiments  on  the  spectrum,  436.  On  diffrac- 
tion and  interference,  740. 

FRESNEL,  his  optical  discoveries  and  researches,  passim.  His 
theory  of  transverse  vibrations,  976.  Of  the  diffracted  fringes 
in  shadows,  718,  His  theorem  for  the  resultant  of  two  inter- 
fering rays,  613.  His  analysis  of  the  colours  seen  through  a 
minute  circular  aperture,  731.  His  experiments  on  the  inter- 
ference of  polarized  rays,  954.  957.  His  laws  of  reflexion  of 
polarized  light,  8.52.  His  theory  of  double  refraction  in 
uniaxal  crystals,  989.  In  biaxal,  997.  His  theory  of  circular 
polarization,  1047. 

Fringes  diffracted,  their  theory,  718.  Their  displacement  by 
interposition  of  a  transparent  plate,  737.  Exterior,  706.  In- 
terior, 726.  Coloured,  seen  between  a  prism  and  a  plane  glass, 
RH.  Between  thick  parallel  plates,  688.  Between  glass  films, 
69.5.  Produced  by  heating  a  glass  plate,  1099. 

Glass,  flint,  and  crovm.  Refractive  and  dispersive  indices  of 
their  varieties.  See  Tables,  Art.  1 1 16. 1120.  Heated,  pressed, 
or  bent,  its  phenomena,  1086.  1090.  1095.  Unannealed, 
110J. 

Heat,  its  effect  in  changing  colours  of  bodies,  504.  Of  crystals, 
unequal  on  the  two  pencils,  1077.  Effects  of  unequal  heat  on 
glass,  1083.  1095.  On  crystallized  bodies,  their  forms  and 
double  refractions,  1 109. 

Hemitrofmm,  remarkable  cases  of,  detected  by  polarized  light, 
1127,^  seq. 

Homogeneous  light,  ils  properties,  600.  Purification,  412.  In- 
sulation, 503.  Lengths  of  undulations  for  its  several  species, 
576. 

Humours  of  the  eye,  350.  854. 

HUYGENS,  his  law  of  velocity  of  the  extraordinary  ray  in  Iceland 
spar,  787.  His  construction  for  law  of  extraordinary  refrac- 
tion, 806.  Extended  to  biaxal  crystals,  1011. 

Ice/and  spar,  phenomena  of,  polarization  and  double  refraction 
exhibited  by,  879,  &c.  Dichroism  of,  106S.  Pyrometrical 
properties  of,  1110.  Interrupted,  phenomena  of,  1080. 

Jdiocyclophanous  crystals,  1081. 

Illumination,  formula  for  its  intensity,  44.  47.  Of  the  polarized 
rings  at  different  points  of  their  peripheries,  1071. 

Images,  319.  Form  of,  320.  Rule  to  find  their  places,  344. 
Brightness  of,  349.  Formed  within  the  eye,  357. 

Incommensurability  of  coloured  spaces  in  the  spectrum,  441. 

Index  of  refraction,  how  determined,  206.  213.  Wollaston's 
method,  562.  Fraunhofer's,  436.  Arago  and  Fresnel's,  739. 
By  polarizing  angle,  843.  Table  of  its  values,  1116. 

Index  of  transparency,  486. 

Inflexion  of  light,  Newton's  doctrine  of,  713. 

Intensity  of  light,  its  law  of  diminution,  18.  Its  measure  in  the 
undulatory  doctrine,  578.  Reflected  perpendicularly,  calcu- 
lated, 592. 

Intensity  of  a  polarized  beam  reflected  in  any  plane,  852.  Of 
natural  lights  when  so  reflected,  857.  592.  Of  the  comple- 
mentary pencils  produced  by  double  refraction,  873.  987.  Of 
the  polarized  rings  at  any  points  of  their  periphery,  1071. 

Interferences  of  rays,  59fi,  et  seq.  General  investigation  of,  618. 
Young's  fundamental  experiment,  726.  Of  polarized  rays,  946, 
et  seq. 

Interrupting  films,  their  phenomena,  1078. 

Irrailialion,  697. 

Isochromatic  lines,  906. 

Jellies,  polarization  of  light  produced  by,  when  dilated  or  com- 
pressed, 1094. 

Least  action,  principle  of  its  use  in  optical  investigations,  536, 
Its  general  application,  540.  Its  equivalent  in  the  undulatory 
doctrine,  588.  Application  to  the  theory  of  uniaxal  crystals. 
790. 

Lemniscates,  polarized,  surrounding  the  axes  of  biaxal  crystals, 
902.  See  Rings,  Tints,  &c. 

Lenses,  259.     Aplanatic,  304.     "  Crossed,''  305. 

Liquids,  rotatory,  phenomena  produced  by,  1045. 

Longitudinal  and  lateral  aberration,  283. 


MALUS,  his  theory  of  double  refraction,  196.  805.     His  discovery       Indet. 
of  polarization  of  light  by  reflexion,  822.  ^•N^^> 

MAYER,  his  hypothesis  of  three  primary  colours,  409. 

Media,  dichromatic,  499. 

Metals,  their  action  in  polarizing  light  hy  reflexion,  84S. 

Microscopes,  309.  389. 

MITSCHERLICH,  M.,  his  researches  on  the  effects  of  heat  on 
crystals,  1109.  i 

jVadifications  of  light,  80. 

Molecules,  luminous,  their  tenuity,  543.  Their  motion  on  chang- 
ing media  investigated,  528. 

NEWTON,  his  theory  of  light,  526.  Doctrine  of  inflexion  and 
deflexion,  713.  Theory  of  colours  of  natural  bodies,  1134. 
Of  the  size  of  their  particles,  1145. 

Object  glass,  achromatic,  its  theory,  459.  et  seq.  General  equa- 
tion for  destroying  its  aberrations,  465.  Aplanatic,  its  con- 
struction, 468.  470,  &c.  With  separated  lenses,  479.  With 
fluid  lenses,  474. 

Oblique  incidence,  its  effect  on  the  colours  of  thin  plates.  6*9. 
657.  Pencils,  their  foci,  321 .  328.  Reflexion  from  water,  553. 

Opacity,  its  cause  on  Newton's  doctrine,  1 138. 

Origin,  of  a  ray  in  the  undulatory  doctrine,  607.  609. 

Periodical  colours,  635,  et  seq. 

Periodicity,  law  of,  906. 

Phaie  of  an  undulation,  604. 

Photometers,  57.     Photometry,  17,  etseq. 

Piles  of  transparent  plates,  their  phenomena  in  polarized  light, 
869. 

Plagiedral  quartz,  its  rotatory  phenomena,  1042. 

Plane  of  polarization,  828.  Its  change  by  reflexion,  860.  Its 
apparent  rotation  in  quanz,  &c.  1039.  Its  oscillations,  928. 

Plates,  thin,  tl.eir  colours,  633, etseq.  Thick,  dilto,  676.  Mixed, 
ditto,  696.  Crystallized,  their  phenomena,  936.  (See  Rings.) 
Crossed,  9JS,  939.  Superposition  of,  9 10. 

PotssoN,  M.,  his  theorem  for  the  illumination  of  the  shadow  of 
a  small  circular  disc,  and  the  colours  seen  through  a  minute 
aperture,  734.  His  investigation  of  the  intensity  of  reflected 
light,  592. 

Polarization  of  light  generally,  814,  et  seq.  Modes  of  effecting, 
819.  Characters  of  a  polarized  ray,  820.  By  reflexion,  821, 
et  seq.  Partial,  847.  By  several  reflexions  in  one  plane,  848. 
By  refraction,  863.  By  several  oblique  transmissions,  863. 
866.  By  double  refraction,  873.  Movable,  Blot's  doctrine  of, 
928.  Explained  on  the  undulatory  doctrine,  993.  Its  princi- 
ples applied  to  the  phenomena  of  biaxal  crystals,  1071. 
Circular,  its  characters,  1049.  How  effected,  1052.  Plane  of, 
its  position  in  the  interior  of  biaxal  crystals,  1070.  Of  sky 
light,  1143. 

Polarized  rings,  surrounding  the  optic  axes  of  crystals,  mode  of 
viewing,  892,  et  seq.  Their  form  in  general,  902.  In  uniaxal 
crystals,  911.  Dependence  of  their  tints  on  law  of  interferences, 
912.  Primary  and  complementary  sets  of,  926.  Explained  on 
hypothesis  of  movable  polarization,  931.  On  undulatory 
hypothesis,  969. 

Polarizing  angle,  Brewster's  law  for  determining.  831.  Its  use  as 
a  physical  character,  1 123 

Polarizing  energy,  a  physical  character,  1126. 
Poles  of  lemniscates,  902.     Virtual,  in  biaxal  crystals,  924. 
Power  of  a  lens,  262.     Of  a  system  of  spherical  surfaces,  270. 
Magnifying,  374.     Superposition  of  powers,  law  of  in  lenses. 
268. 

Pressure,  ils  effect  in  communicating  the  property  of  polariza- 
tion, 1087. 
Principle  of  least  action  applied  to  double  refraction,  790.    Of 

swiftest  propagation,  588. 

Prism,  formulae  for  refraction  through,  198,  etteq.     Of  variable 
refracting  angle,  431,  432.     Analysis  of   light  by,  397.    Tele- 
scopes composed  of  prisms,  453.    Coloured  bow  seen  in,  555. 
Propagation  of  light,  5.     Oersted's  hypothesis  for,  525.     Law  of 

swiftest,  588.     Of  waves  along  canals,  600. 
Punctum  caecum  in  the  eye,  366. 

Quartz,  right  and  left-handed,  1041.  Rotatory  phenomena  in, 
IOS7.  Double  refraction  of  along  its  axis,  1048.  Plagiedral,  it* 
phenomena,  1042. 

Radiation  of  light,  5,  et  seq.  Its  law,  72.  Explained  on  undu- 
lalory  doctrine,  578. 

Ray:,  calorific,  luminous,  and  chemical,  1147.  Similar  and  dis- 
similar, 606.  Their  origins,  607.  Interfering,  their  resultant, 
611.  Polarized,  their  characters,  890. 


L  I  G  -H  T. 


585 


Light. 


RtJIfcting  Jbrcet,  their  intensity,  561.  Distribution,  550,  el  set/. 
/  Reflexion,  law  of,  88.  General  formulae  for,  at  plane  surfaces,  99. 
At  curved  surfaces,  108,  109.  Between  any  system  of  spherical 
surfaces,  301.  Internal  total,  181.  550.  554.  Modification 
impressed  on  light  by  two  such,  1056.  At  common  surface  of 
two  media,  547.  Partial,  explained  on  Newton's  principles, 
544.  Regular  at  rough  or  artificially  polished  surfaces  ex- 
plained, 557.  558.  How  conceived  in  the  undulatory  doctrine, 
584.  At  the  surfaces  of  crystals,  1123.  Of  polarized  light,  its 
laws,  849,  et  seq. 

Refraction,  by  uncrystallized  media,  n\,etseq.  Its  law,  189. 
General  formuUe  for,  at  plane  surfaces,  198.  Through  prisms, 
90S.  2 1 1.  At  curved  surfaces,  220,  et  seq.  At  common  surface 
of  two  media,  189.  Colourless,  a  case  of,  478.  Regular,  at 
artificially  polished  surfaces,  explained,  559.  Account  of  in 
undulatory  theory,  586.  595.  628. 

Refraction,  double,  779,  et  seq.  By  what  bodies  produced,  780. 
Its  law  in  uniaxal  crystals,  785. 800.  Produced  by  rock  crystal 
along  its  axis,  1048.  By  compressed  and  dilated  glass,  1107. 
In  uniayal  crystals,  explained  on  undulatory  doctrine,  989.  In 
biaxa.,  us  general  laws,  101 1.  1014.  Ordinary  and  extraordi- 
nary, relation  of  the  two  pencils,  873. 

Refracting  forces,  their  intensity  and  extent,  561. 

Refractive  power,  intrinsic,  5S5.  Table  of  its  values  in  different 
media,  1118.  Its  connection  with  their  chemical  composition, 
1114. 

Refractive  index,  how  measured,  see  Index.  Table  of  its  values 
for  different  media,  1116.  For  different  homogeneous  rays, 
437. 

Refrantfibilily  of  different  rays.     See  Chromatics,  Colours,  &c. 

Resultant  of  two  interfering  vibrations,  61 1.  Of  rays  oppositely 
polarized,  982. 

Retina,  S.55.     How  affected  by  vibrations  of  ether,  567. 

Rings,  coloured,  seen  between  convex  glasses,  their  colours,  635. 
Breadths,  657.  For  different  homogeneous  rays,  644.  Their 
analysis  and  synthesis,  644,  645.  Transmitted,  658.  Ex- 
plained on  the  undulatory  theory,  660.  On  the  Newtonian, 
655.  Seen  about  the  images  of  stars  in  telescopes,  766.  Seen 
about  the  poles  of  the  optic  axes  in  crystals,  892.  900.  Law  of 
their  intensity  in  different  points  of  their  circumference,  1071. 

Rotatory  phenomena  of  rock  crystal  and  liquids,  1038.  1040. 
Ex  plained  on  the  undulatory  doctrine,  1057. 

SEEBECK,  Dr.,  his  discovery  of  the  rotatory  property  in  liquids, 
i045.  Of  the  effects  of  heat  in  imparting  polarization  to 
glass,  1083. 

Sections,  principal,  of  a  crystallized  plate,  888. 

Soap  bubbles,  colours  reflected  by,  649. 

Solar  light,  its  analysis  by  the  prism,  397.  Its  peculiar  cha- 
racters and  spectrum,  419. 

Spectrum,  prismatic,  397.  Fixed  lines  in,  418  ;  secondary,  442  ; 
tertiary,  446.  Its  distortion  at  extreme  incidences,  450  ; 
subordinate,  452.  Of  first  class,  760  ;  of  second  class,  740  ; 
of  third  class,  761. 

Spheroid  of  double  refraction  in  uniaxal  crystals,  789.  In 
biaxal,  10  IS. 

Spherometer,  1111. 

fitars,  their  spurious  discs  and  rings,  766,  et  seq. 

Statues,  musical  sounds  produced  by  certain,  a  probable  expla- 
nation of,  1 103. 

Strain  of  solids,  ascertained  by  their  polarized  lints,  1090.  Stale 
of,  in  unequally  heated  glass  plates,  1098. 

Sulphate  of  copper  and  potash,  a  singular  property  of,  1 1 1 1 .  Of 
lime,  action  of  heat  in  altering  its  optical  properties,  HIV. 
Of  potash,  singular  structure  of  its  crystals,  1 132. 

Table  of  media  in  their  order  of  action  in  green  light,  <I4S.  Of 
dispersive  powers  of  first  and  second  order  on  a  water  scale, 
447.  Of  the  lengths  of  undulations  of  the  several  homoge- 
neous rays,  575.  756.  Of  the  maxima  and  minima  of  the  ex- 
terior fringes  of  shadows,  720.  Of  colours  seen  by  a  person 


of  defective  vision,  507.  Of  colours  seen  by  diffraction 
through  a  circular  hole,  730.  Of  the  dimensions  of  the 
lemniscates  in  mica,  908.  Of  crystals  whose  optic  axes  differ 
for  different  rays,  (Brewsler,)923.  Of  the  angles  of  rotation  of 
the  several  homogeneous  rays,  1040.  Of  refractive  indices, 
(general,)  1116.  Of  refractive  indices  for  seven  definite  rays, 
(Fraunhofer,)  437.  Of  intrinsic  refractive  powers,  1118.  Of 
dispersive  powers,  (general,)  1120.  Of  angles  between  the 
optic  axes  of  various  crystals,  1124.  Of  polarizing  powers, 
1126. 

Telescopes,  379.  Astronomical,  380.  Galilean,  380.  Hersche- 
lian,  390.  Newtonian,  391.  Prismatic,  453.  Achromatic, 
(see  Achromatic  ) 

Tesselite,  its  singular  structure,  1130,  1131. 

Theories  of  light,  Newtonian,  526.     Undulatory,  563,  etseq. 

Thick  plates,  colours  of,  676,  et  sey.  Explained  on  undulatory 
system,  678. 

Thin  plates,  colours  of,  633,  et  sej.  Newton's  explanation  of 
them,  651. 

Tint,  its  numerical  measure,  906. 

Tints  of  coloured  media,  vary  with  a  change  of  thickness,  495. 
Of  transmitted  rings  expressed  algebraically,  663.  667.  Of 
crystallized  plates,  their  law,  886.  906.  Their  dependence  on 
the  thickness  of  the  plate,  905.  Their  deviation  from  Newton's 
scale,  915.  Singular  succession  of,  exhibited  by  Vesuvian, 
1125.  Of  circular  polarization,  1055. 

Transparency,  on  what  depending,  1 142.  Index  of,  486.  Of 
oiled  paper,  &c.  549. 

Tourmaline,  its  property  of  polarized  light,  817-  Of  absorbing 
one  pencil,  1060. 

Type,  of  the  colour  of  a  medium,  490.  Instanced  in  various 
cases,  498. 

Ultimate  lint  of  an  absorptive  medium,  494. 

Viiannealeil  glass,  its  optical   properties,   1104. 

Undulations  of  ether,  574.  Their  lengths  for  homogeneous  rays, 
575.  736.  Their  phases,  COI.  Amplitudes,  605.  Propaga- 
tion in  spheroidal  surfaces,  804. 

Undulation,  half  an,  allowance  for  cases  when  required,  966" 
672.  717.  Kresnel's  rule  for,  972.  Explained,  a  priori,  983- 

Velocity  of  lighi  9.13.  Of  etherial  undulation,  56 1.  Of  plane 
waves  within  crystals,  1005.  1012.  Of  ordinary  and  extraordi- 
nary ray  on  Huygenian  hypothesis,  787.  Of  luminiferous 
waves  and  of  rays,  distinguished,  813. 

I'esuvian,  its  remarkable  inverted  scale  of  lints,  1125. 

Vibrations  of  ether,  rectilinear,  their  laws,  569.  Resultant  of 
two  interfering,  61 1.  Their  composition  and  resolution,  620  . 
particular  cases,  621.  Elliptic,  621.  Circular,  627. 

Vibration,  ils  effect  in  imparting  polarizing  power  to  glass,  1093. 

l-'mon,  350.     Single,  with  two   eyes,  361.     Double,  361.  363. 
Restoration  of  at  an  advanced  age,  360.     Through  lenses,  &c. 
376.     Of  persons  who  see  only  two  colours,  507.      Oblique 
through  refracting  or  reflecting  surfaces,  341. 
Visual  angle,  376. 

Water,  its  indices  of  refraction  for  seven  definite  rays,  437.     Its 

spectrum,  438. 
Wove*  of  light  explained,  573.     Secondary,  583;  their  mutual 

destruction,  628.     Transmitted  through  apertures,  631.     Plane, 

their   velocity   and    direction  in    crystals  investigated,   1012. 

Curved,  the  general  equation  of  their  surface  in  biaxal  crystals, 

1013. 
WOI.I.ASTON.  Dr.,  his  determinations  of  refractive  indices,  1115. 

His  researches  on  double  refraction  of  Iceland  spar,  780.     On 

the  chemical  rays,  1147.     Discovery  of  the  fixed  lines  in  the 

spectrum,  418. 

YOUNG,  Dr.,  his  law  of  interference.  See  Interferences.  Ilia 
analogy  between  the  vibrations  of  ether,  and  those  of  a  tended 
cord,  977.  His  optical  dis<  overiesand  investigations. JM.WKI 


Index. 


VOL.  IV. 


586  LIGHT. 


ERRATA  ET  CORRIGENDA. 

N  B.  The  reader  is  requested  to  correct  in  advance  the  following  Errata,  and  to  strike  out  the  passages  here  referrw  \o 

Page,  Line.  Error.                                               Correction. 

341,  SO,  existences,  existence. 

do.  31,  moie,  most. 

347,  2,  line,  sine. 

349,  26,  as  the  sun's  surface,  at  the  sun's  surface 

389,  5  from  bott.       -  -^-  (  --  - 


-  -  - 

2  ft  I  2  p.  I, 

399,  47,  axis,  axes. 

400,  15,  act,  art. 

401,  15,  PE,  PC. 

402,  30,  R,  UOp,  P,  POp. 
Ho.  35,  p,  Q. 

410,  22  from  bott.      dele  "  see  Micrometer,  in  a  subsequent  part  of  this  Article.' 

414,  44,  dele  "  by  the  writer  of  these  pages." 

415,  3  from  bott.       by  water,  into  water. 

420,  30,  spectra  of  distortion,  subordinate  spectra. 

428,  17,  secondary,  second. 

431,  S3,  Rr,  Vti,  R  R',  V  V. 

do.  36,  R  N  V,  R'  N  V. 

434,  27,  from  experiments,  from  other  experiments. 

454,  28,  P  B,  P  Q,  or  A  B,  P  B  -  P  Q,  or  A  B. 

461,  11   from  bott.       two  vibrations,  two  rectangular  vibrations. 

476.  14  from  bott.      dele  all  that  relates  to  the  fringes  on  the  wings  of  the  Papilla  Idas,  being  founded 

on  a  mistake. 

480,  32,  limits,  limit. 

509,  38,  fails,  falls. 

521,  1,  produce,  to  produce. 

524,        margin,  Art.  925,  polarization,  depolarization. 

528,  22,  positive  class,  attractive  class. 

531,  16,  add  as  foliates:  —  With  respect  to  this  third  law,  however,  it  must  be  confessed  that  it  appears 

to  require  a  stricter  examination,  as,  if  admitted  in  its  full  extent,  it  seems 
to  controvert  the  fundamental  principles  of  the  doctrine  of  interference 

564,  19,     dele  what  is  said  about  the  nodal  lines. 

560.  S2  from  bott.      after  disprove  it,  insert  as  follows:   Instead   of  the   expression  (A,)  Art.   10T3,  w« 

might  otherwise  assume 

T  =  (Y  .  cos  2  0s  +  B  .  sin  2  if-)  .  (y  .  cos  x9  +  A  .  sin  ys), 
and  determining  the  coefficients  accordingly,  obtain  another  expression  for  the  tint. 


L  I  G  H  T 


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